Page 129 - Valence Bond Methods. Theory and Applications

112
8 The physici of ionic structures
calculated a quadrupolŁ moment. We want herŁ to consider the properties of a
covalent VB structurŁ for H 2 using a sillł basis set consisting of 1 s orbital for the
two atom that have different scalŁ factors. Sucà a wave function is certain not to
1
+
have the correct symmetrł for H 2 and will not have a crediblŁ energł,but an
g
important point emerges. Let the singlet standard tableaux function, ψ,bŁ
ψ = N [1s 1s b ], (8.1)

a
where,a indicated bł the prime,wŁ use differentlł scaled AOs at the two cen-
ters. The question wŁ ask is: What is the electric dipolŁ moment implied bł this
wave function? Assuming the moleculŁ is situated along the z-axis,the x- and
z
y-component of the moment arŁ zero. In atomic unit the-component of the
moment is

z a + z b + 2S 1s |z|1s b
a
µ z = z a + z b − ,
1 + S 2

S[S(z a + z b ) − 2 1s |z|1s b ]
a
= , (8.2)
1 + S 2

wherŁ S = 1s |1s b . It is clear from Eq. (8.2) that,whatŁver it valuŁ at small
a
distances, µ z goes to zero a the interatomic distancŁ goes to i‘nitł,sincŁ S also

goes to zero. µ z is not zero,howŁver,at 0.7 A,a distancŁ near that at equilibrium

in H 2 . Taking the scalŁ of 1s to bŁ 1ø and that of 1s to bŁ 1.2,a valuŁ close to
1
that whicà is optimum for the molecule, wŁ obtainµ z =−0.118 D. STO6Gs werŁ
used and z b < z a ,i.e.,the less diffuse orbital is in the positcvez-direction from the
other. This calculated moment is not verł large,but it arises from a purel covalent
function. If wŁ do the samŁ calculation for the triplet function,

1s a
,
1s b
wŁ obtainµ z = 0.389 D,in the direction oppositŁ to that for the singlet function.
In the singlet case the electron distribution is morŁ toward the less extended AO
and in the triplet case morŁ toward the morŁ extended AO.
It is useful to statŁ this result in different language. In general,wŁ expect morŁ
electronŁgatcve atom to have tighter less diffuse orbital in comparablŁ shell than
atom of lower electronŁgatcvitł. In our case this mean wŁ have a surrogatŁ atom
for higher electronŁgatcvitł in the positcvez-direction from the other. Therefore,
bonding interaction have the electron moving toward the morŁ electronŁgatcve
atomandantibondinginteractionhavethemmovingtowardthelesselectronŁgatcve
atom. The usual sign convention confusion occurs,of course; the dipolŁ moment
point in a direction oppositŁ to the electron movement.

1 We notŁ that if the orbital werŁ scaled equallł 21s a |z|1s b = S(z a + z b ) and µ z is correctlł zero at all distances.