Physical Chemistry Third Edition

870 21 The Electronic Structure of Polyatomic Molecules

(3) of determinants in Appendix B, the value of the Slater determinant is unchanged

except for being multiplied by a constant, so that the antisymmetrized wave function

is not changed. These new orbitals are

ψσ 2 ′c 1 (0. 17095 ψ 1 sBe− 0. 40852 ψ 2 sBe+ 0. 44243 ψ 2 pzBe

− 0. 00907 ψ 1 sHa− 0. 89159 ψ 1 sHb) (21.1-1)

and

ψσ 3 ′c 2 (0. 17095 ψ 1 sBe− 0. 40852 ψ 2 sBe− 0. 44243 ψ 2 pzBe

− 0. 89159 ψ 1 sHa− 0. 00907 ψ 1 sHb) (21.1-2)

The first of these orbitals contains almost no contribution from hydrogen a and the

second contains almost no contribution from hydrogen b. They correspond very nearly

to chemical bonds that consist of pairs of electrons shared by two nuclei. These orbitals

give the same wave function as the canonical orbitals after antisymmetrization.

The coefficients of the 2sBe and the 2pzBe atomic orbitals in our new LCAOMOs

ψσ 2 ′andψσ 3 ′are roughly equal in magnitude and the coefficients of the 1sBe orbital

are quite small. This is similar to the situation in the LiH molecule that we dis-

cussed earlier, and we can to an adequate approximation introduce the 2sphybrid

orbitals. Since Maclagan and Schnuelle used Slater-type 2sorbitals, which are positive

everywhere, the expressions for the hybrid orbitals in Eqs. (20.4-6) and (20.4-7) are

replaced by

ψ 2 sp(1)

√

1 /2(ψ 2 s+ψ 2 pz) (21.1-3)

and

ψ 2 sp(2)

√

1 /2(ψ 2 s−ψ 2 pz) (21.1-4)

We now write the approximate LCAOMOs

ψ 2 ′≈c 1 (− 0. 40852 ψ 2 sBe+ 0. 44243 ψ 2 pzBe− 0. 89159 ψ 1 sHb)

≈c 1 (− 0. 42

√

2 ψ 2 sp(2)Be− 0. 90 ψ 1 sHb) (21.1-5)

≈c 1 (− 0. 59 ψ 2 sp(2)Be− 0. 90 ψ 1 sHb)

and

ψ 3 ′≈c 2 (− 0. 40852 ψ 2 sBe− 0. 44243 ψ 2 pzBe− 0. 89159 ψ 1 sHa)

≈c 2 (− 0. 42

√

2 ψ 2 sp(1)Be− 0. 89 ψ 1 sHa) (21.1-6)

≈c 2 (− 0. 59 ψ 2 sp(1)Be− 0. 89 ψ 1 sHa)

We now have two localized bonding orbitals, each one of which is made with a 2sp

hybrid orbital on the Be atom and a 1shydrogen orbital, and is roughly the same

as one of the orbitals in Eqs. (21.1-1) and (21.1-2). Each of these space orbitals is

occupied by two electrons, making two single bonds between the Be atom and the two

H atoms.

The LCAOMOs conform to the same pattern that we observed with LiH: The 1sBe

orbital corresponds to a much lower energy than the other atomic orbitals, and is

a nonbonding orbital. The two bonding LCAOMOs correspond to significant over-

lap between a hybrid orbital and a hydrogen 1sorbital. The hybrid orbital and the