Physical Chemistry Third Edition

20.2 LCAOMOs. Approximate Molecular Orbitals That Are Linear Combinations of Atomic Orbitals 833

a.Find the resulting point when the product of two

operatorsC 2 (z)C 2 (x) operates on

⎡

⎣

a

b

c

⎤

⎦. Give the

single operator equivalent to the product of the two

operators.

b.Repeat part a for the operators in the other order.

c.Do the two operators commute?

20.2 LCAOMOs. Approximate Molecular

Orbitals That Are Linear Combinations

of Atomic Orbitals

The exact Born–Oppenheimer orbitals for H+ 2 are expressed in an unfamiliar

coordinate system and we did not explicitly display them. It will be convenient to

have some easily expressed approximate molecular orbitals that resemble the correct

molecular orbitals. We definemolecular orbitals that are linear combinations of atomic

orbitals, abbreviated LCAOMOs. Iff 1 ,f 2 ,f 3 ,...are a set of basis functions, then a

linear combinationof these functions is written as in Eq. (16.3-34):

gc 1 f 1 +c 2 f 2 +c 3 f 3 + ··· 

∑∞

j 1

cjfj (20.2-1)

where theccoefficients are constants. If the linear combination can be an exact

representation of any function that obeys the same boundary conditions as the basis set,

the basis set is said to be acomplete set. A complete basis set ordinarily has infinitely

many members. We will not attempt to use a complete set of basis functions, but will

begin with a basis set consisting of two atomic orbitals.

We seek approximate LCAOMO representations for theψ 1 σgandψ 2 σ∗umolecular

orbitals. We denote one nucleus as A and the other as B. We letrAbe the distance from

nucleus A to the electron, and letrBbe the distance from nucleus B to the electron.

As our first basis set we take a hydrogen-like 1sorbital withrAas its independent

variable and a hydrogen-like 1sorbital withrBas its independent variable. We use the

abbreviations:

ψ 1 sAψ 1 s(rA) (20.2-2a)

ψ 1 sBψ 1 s(rB) (20.2-2b)

The orbitalψ 1 sAhas its orbital region centered at nucleus A and the orbitalψ 1 sBhas

its orbital region centered at nucleus B. We now form molecular orbitals that are linear

combinations of these two basis functions:

ψMOcAψ 1 sA+cBψ 1 sB (20.2-3)

We can obtain two independent linear combinations from two independent basis func-

tions, and we seek two that are approximations to the exact Born–Oppenheimer

molecular orbitalsψ 1 σgandψ 2 σ∗u.

One way to find the appropriate values ofcAandcBis to regardψMOas a variation

trial function and to minimize the energy as a function ofcAandcB. We do not present