MOLECULAR ORBITAL THEORY

Structure, Bonding, and Properties

Rule 4: Linking of polyhedra having different cations

1.9. MOLECULAR ORBITAL THEORY

In Fig. 1.15 the additional stabilization by the Jahn-Teller effect has not been taken into account. Its inclusion brings the point for the (distorted) octahedral coordination for Cu2+ further down, thus rendering this arrangement more favorable.

40

^{TABLE 1.9.} COMMON COORDINATION POLYHEDRA FOR TRANSITION METAL COMPOUNDS

CHAPTER 1 Coordination Electron

Polyhedron Number Configuration Central Atom Examples

Linear arrangement 2 dlO Cu(l), Ag(I), CU20, HgOa

Au(l), Hg(II)

Triangle 3 dlO Cu(l), Ag(l), Cu(CN)/-,

Au(l), Hg(II) Ag₂C1_53-

Square 4 dB Ni(II), Pd(II), Ni(CN)/-, PdC12a ,

Pt(II), Au(II) AuCl4 -, Pt(NH3hCI2

Tetrahedron 4 dO Ti(IV), V(V), TiC4, V04^3-,

Cr(VI), Cr03a, Cr04^2-,

Mo(VI), M004^2-

Mn(VII), Mn207, Re04 -, Re(VII),

Ru(VIII), RU04,

Os(VIII) OS04

d^l V(lV), Cr(V), VC4, cr04 ^3-,

Mn(VI), Mn04^2-,

Ru(VII) Ru04-

d⁵ Mn(II), MnBrl-,

Fe(III) Fe2Cl6

d⁶ Fe(II) FeC14^2-

d⁷ Co(II) CoC14^2-

dB _{Ni(II) NiCl/-}

d⁹ Cu(II) CUC42-b

d^lo Ni(O), Cu(l), Ni(CO)4, CU20, Zn(II), Hg(II) Zn(CN)/-, HgI/-

Square pyramid 5 dO Ti(IV), V(V), TiOCI4^2-, VoF4-,

Nb(V), NbSCl4 -,

Mo(VI), MoNCI4-,

W(VI) WNCI4-

d^l V(IV), VO(NCS)/-,

Cr(V), CrOCl4 -,

Mo(V), MoOCI4-

W(V), WCCI4-,

Re(VI) ReOC4

d² Os(VI) OsNCl4 -

d⁴ Mn(III), MnCI_52-,

Re(III) Re2Cl8

d⁷ Co(ll) Co(CN)s3-

Trigonal bipyramid 5 d² V(lV) VCI3(NMe3h

d⁸ Fe(O) Fe(CO)s

Octahedron 6 nearly all;

rarely Pd(II), Pt(II), Au(III), Cu(l)

a Endless chain.

b J ahn-Teller distorted.

The wave function of an electron describes the amplitude of a vibrating chord as a function of the position x, y, z. The opposite direction of the motion of the chord on the two sides of a vibration node is expressed by opposite signs of the wave function.

Similarly, the wave function of an electron has opposite signs on the two sides of a nodal surface. The wave function is a function of the site r referred to a coordinate system

which has its origin in the center of the atomic nucleus. Wave functions for the orbitals of molecules are calculated by linear combinations of all wave functions of all atoms involved. The total number of orbitals remains unaltered; in other words, the total number of contributing atomic orbitals must be equal to the number of molecular orbitals.

Furthermore, certain conditions have to be obeyed in the calculation, including linear independence of the molecular orbital functions and normalization. In this book we will designate wave functions of atoms by <Pj and wave function of molecules by \jI. We obtain the wave functions of an Hz molecule by linear combination of the Is functions <PI and <Pz of the two hydrogen atoms:

and

In comparison to an H atom, electrons with wave function \jI1 have lower energy, and those with \jIz have higher energy. Thus, the two electrons prefer to "occupy" the molecular orbital \jIj.

To calculate the wave functions for the bonds between two atoms of different elements, the new functions of the bonded atoms should be a linear combination of the bonding atoms with coefficients CI and Cz;

\jIj = cI <PI

+

c2<P2

\jI2 = c2<P2 - CI <P2

(1.9) (1.10) The probability of finding an electron at a site (x, y, z) is given by 1\jI12. Integrating over all space, the probability must be equal to 1:

J

^{dr 1\jII2}

⁼ J

dr ICI <PI

+

c2<P21 2

= c1 + c~ +

^{2cI C2S12}

=

1 (1.11 ) where S12 is the overlap integral between <PI and <Pz. The term 2cI c2S12 is the overlap population and describes the electronic interaction between the atoms. The contributions

ct

^{and ~} can be assigned to the atoms 1 and 2, respectively. Equation (1.11) is fulfilled if

CI

~ 1 and ~ ~ O. That means the electron is localized mainly at atom 1 and the overlap population is approximately zero. This is the situation of a minor electronic interaction, either because the corresponding orbitals are too far apart or because they differ considerably in energy. Therefore, such an electron does not contribute to bonding. For

\jI1 the overlap population 2cI c2S12 is positive, and the electron is bonding. For \jIz it is negative and the electron is antibonding. The sum of the values 2cI c2S12 of all occupied orbitals of the molecule, called Mulliken overlap population, is a measure of the bond strength or bond order (BO), with BO = (112) [(number of bonding electrons) - (number of antibonding electrons)]. Any other orbitals can also be combined to bonding, antibonding, or nonbonding molecular orbitals. Nonbonding are those orbitals for which bonding and antibonding components cancel with each other. Some possibilities are shown in Fig. 1.16. Note the signs of the wave function. A bonding molecular orbital having no nodal surface is a cr orbital; if it has one nodal plane parallel to the connecting line between the atomic centers it is a 1t orbital, and with two such nodal planes it is a 8 orbital. Antibonding orbitals usually are designated by an asterisk.

41

STRUCTURE, BONDING, AND PROPERTIES

42

CHAPTER 1

p 0* p

d

P 1t P P 1t* P d ^1t p

Figure 1.16. Some combinations of atomic wave functions in molecular orbitals. Asterisks mark antibonding orbitals.

1.9.2. HYBRIDIZATION

A univalent atom has one orbital available for bonding. But atoms with multiple valences, for example 2, 3, 4, or more, must form bonds by using at least two orbitals. An oxygen atom has two half-filled orbitals in which each one only can add one electron, so it has valence of 2 and forms a single bond by the overlap of these with the orbitals of one or two other atoms. Because the maximum overlap of the electron wave function,

2CIC2S12' has lower energy, it is a stable bonding condition. The two atoms that bonded with an oxygen atom should have almost perpendicular relation due to the p valence orbitals of the oxygen atom. However, the observed bonding angle is not 90° due to the repulsive interaction of the electrons in the orbitals (for example in water it is 104°27').

By the same token, three-valent nitrogen should have three half-filled orbitals, and four- valent carbon should have four half-filled orbitals. We now use methane as an example to describe the molecular orbital theory.

For understanding the structure of a methane it is better to calculate the orbitals for a methane molecule. A methane has four hydrogen and one carbon, therefore the four Is wave functions of the four hydrogen atoms and the wave functions 2s, 2px, 2py, and 2pz of the carbon atom are combined to give eight wave functions, four of which are bonding and four of which are antibonding. The four bonding wave functions are

(1.12)

~1 to ~4 are the wave functions of CH4 molecule, S,Px,Py' and pz designate the wave functions of the C atom, and <l>HI' <l>H2' • •• correspond to the H atoms. Among the coefficients Cl' C2' and C3' only c3 is negligible.

Based on these combinations we can get the electron distribution in a methane molecule. If we focus on individual atom to see how the electrons distribute, it can be found that for carbon atom the sand p original orbitals are mixed to form a fraction of the wave function of methane. This is called the mixed s and p orbitals or hybrid orbitals of carbon. These hybrid orbitals will combine with the wave function of hydrogen atoms to form the molecular orbitals of the methane. The spatial orientations of the orbitals, where the electrons distribute, correspond to the orientations of the bonds of the molecule formed. Therefore, the molecular orbitals can be obtained by hybridization of atomic orbitals. Instead of calculating the molecular orbitals of the methane molecule in one step according to the Schrodinger equation, we can do it in two steps. First, only the wave functions of the C atoms are combined to Sp3 hybrid orbitals:

<1>1

=!

^(s

+

^Px

+

^Py

+

^pz)

<1>2

= !

^(s

+

^Px - Py - pz)

<1>3

= !

^{(s - Px}

+

^{Py - pz)}

<1>4

= !

^{(s - Px - Py}

+

^pz)

(1.13)

The functions <1>1 to <1>4 correspond to orbitals having preferential directionality oriented toward the vertices of a circumscribed tetrahedron which has a carbon atom at the center.

Their combinations with the wave functions of the four hydrogen atoms placed in these vertices yield the following functions if the insignificant coefficient C3 is neglected:

~I

=

Cl <1>1

+

c2<1>Hl

~2

=

Cl <1>2

+

C2<1>H2

~3

=

CI<I>3

+

C2<1>H3

~4

=

Cl <1>4

+

C2<1>H4

(1.14)

~1 corresponds to a bonding orbit that essentially involves the interaction of the C atom with the first H atom; its charge density

1"'11

² is concentrated in the region between these two atoms, forming the C-H bond. To be more exact, every bond is a "multicenter bond" with contributions of the wave functions of all atoms due to the charge concentration in the region between the two atoms and the contributions from XH2, XH3, and XH4. The hybridization is a simple and clear description of bonding.

Different hybridization functions are responsible for molecules with different structures. An infinite number of hybridization functions can be formulated by linear combinations of s and P orbitals:

(1.15) The coefficients must be normalized,

IAil2 + IBil2 + IC

i

l

²

+ IDil2

= 1. The fraction contributed by the Px orbital, for example, to the <l>i orbit is I ~i

12.

Their values determine

43

STRUCTURE, BONDING, AND PROPERTIES

44

CHAPTER 1

the preferential directions of the hybridized orbitals. For example, the functions

<PI

=

0.894s

+

^0.447(px

+

^Py

+

^pz)

<P2

=

0.22s

+

^{0.975(px -} Py - pz)

<P3

=

^0.22s

+

^{0.975( -Px}

+

^{Py - pz)}

<P4 = 0.22s

+

0.975( -Px - Py

+

^pz)

(1.16)

<PI defines an orbital having contributions of 80% (0.894²)s and 20%p, and the orbitals

<P2, <P3 and <P4, each with contributions of 5%s and 95%p. They are adequate to calculate the wave functions for a molecule AX3 that has a lone electron pair (<PI) with a large s contribution and two bonds with larger p orbital contributions, i.e., Sp2 hybridization. The corresponding bond angles are 96.so.

To obtain the values of the coefficients \ti' ~i' Yi' and ()i so that the bond energy is maximized and the correct molecular geometry is produced; also the mutual interactions between the electrons must be considered. This requires a great deal of computational expenditure. However, in a qualitative manner the interactions can be estimated rather well using the valence shell electron-pair repulsion theory.