V)'          ho€e(c«P&j   14-,     a. cM^lirilu K*t>\*UAU	i        1 '
		---—-U-1__A	
			
		f. HO-LGAO                   0 =   "a^- + ^	o _i
			Ö
			
			
			
		ô	
			
		^v        o                 A, J_J	
			
			
			
			
		Euev^vc ^0 uXyil 2*oe<>cü; Wudt^ CavC^	>t="
			o ,
			
			^yjpt^i^___
		------- y	
		' »--	
			>
			
			
			
			
	n*l  \ 0	---.
1		
	V   ^                        •                          .      A                   V                 j           1 *	-» ■» v^,----
		
	-toaiiAulíU-  iiW.=   CA£   U -v £ ŕ U       d \	\ + eeU sa M as
		
		S ^>,.,.
		
	0 ^	
	i —*   >        \          /\o         1       z\      i*) — i	
		
	t              Q ^	
	0	
		
		
	U.„=  \ As. U 4s A'l7	
	i	
		
		
	0	
		
	-	
		
		
		
		
		
		
		
TVemk.
<2u
Or H
FIGURE 12-7
Surface plots of (a) the o-,ls and (6) the rruls wavefunctions of the Hj, together with (c) and (<*), the corresponding probability densities (o-,1j)2 and (a-Jsf. The <t,1j and (cr,ls)2.plots look very similar in that both indicate a buildup of the function between the two nuclei. In the case of (o- li)1, this buildup may be interpreted as bonding of the two nuclei. The o-„1j and (o-„ls)' plots show no buildup between the nuclei; in fact, there is a node between the two parts of the functions and a buildup outside the internuclear region. Thus o-u1j describes an antibonding state.
■Riot*;    fVftute i_. p'.c'ci^  &6**«tew Rl*vi*j THE ELECTRONIC STRUCTURE OF LINEAR MOLECULES 349
«1 -(0,1^-i.        + 1,2)
go
h=(°uls)2-\(ls\ +b|)
(6)
flGURE 12-8
differences in probability densities between (a) the bonding o-gLs function and (6) the antibonding ;<Th1j function of according to Eq. (12-34). The two upward-pointing peaks of (&) represent a Igjldup of probability density outside the internuclear region; the sag between these peaks represents a decrease in probability density between the nuclei. Note that this decrease and increase are relative to the density due to two noninteracting Is2 atomic probability densities. The two downward-pointing peaks of (a) represent a decrease of probability density outside the internuclear region, and the mound between these indicates an increase of probability density between the nuclei.
Section 13.7     MO Configurations of Homonuclear Diatomic Molecules 401
TABLE 13.2 Properties of Homonuclear Diatomic Molecules in Their Ground Electronic States
Molecule	Ground Term	Bond Order	iVeV	RJA	vjcm 1
H2	zg iv +	1 2 1	2.79 4.75	1.06 0.741	2322 4403
'He;	2S + w	1 2	2.5	1.08	1698
He2 Li2 Be2 B2	^g ^g ^g Zg ^g	0 1 0 1 2	0.0009 1.07 0.10 3.1 6.3	3.0 2.67 2.45 1.59 1.24	351.4 1051 1855
	22 + ^g	2,	8.85	1.12	2207
N2	^g	3	9.91	1.10	2358
o\	2n g	2\	6.78	1.12	1905
o2 F2 Ne2		2 1 0	5.21 1.66x 0.0036	1.2i , 1.41 3.1	1580 w 14
Data from K. P. Huber and G. Herzberg, Constants of Diatomic Molecules (vqI. IV of Molecular Spectra and Molecular Structure), Van Nostrarid Reinhold, 1979; and (for Be2) V. E. Bondybey, Chem. Phys. Lett, 109,463 (1984).
3.6. Conclusions
It is useful to collect together the main results associated with the interaction of two atomic orbitals located on two centers.
(i) Orbitals can only interact if their overlap integral is non-zero.
(ii) The interaction of two AOs leads to the formation of two MOs, one bonding and one antibonding.
(iii) The bonding MO is more stable than the lower energy orbital of the starting AOs.
(iv) The antibonding MO is less stable than the higher energy orbital of the starting AOs.
(v) The bonding orbital is stabilized less than the antibonding orbital is destabilized.
(vi) If the AOs are degenerate, their interaction is proportional to their overlap integral, S.
(vii) If the AOs are non-degenerate, their interaction is proportional to S2/Ae, where Ae is the energy separation between the AOs. In this case the bonding MO is preferentially localized on the atom with the deeper lying AO, usually the more electronegative atom. The antibonding MO is preferentially localized on the atom which holds the higher energy AO.
TAB LE 13.2 Properties of Homonuclear Diatomic Molecules in Their Ground Electronic States
Molecule	Ground Term	Bond Order	DjeV	RJA	vjcm
H$		1 2	2.79	1.06	2322
H2		1	4.75	0.741	4403
He^	M	1 2	2.5	1.08	1698
He2		0	0.0009	3.0	
Li2		1	1.07	2.67	351.4
Be2		0	0.10	2.45	
B2	3y-*g ly + ^g	1	3.1	1.59	1051
Q		2	6.3	1.24	1855
N+2	%+	2§	8.85	1.12	2207
N2		3	9.91	1.10	2358
	2n		■ 6.78	1.12	1905
o2	3y-^g 15 + *g ly + ^8	2	5.21	1.21 .	1580
F2 Ne^		1 0	1.66 s 0.0036	1.41 3.1	8J2 14
Data from K. P. Huber and G. Herzberg, Constants of Diatomic Molecules (vqI. IV of Molecular Spectra and Molecular Structure),Van Nostrand Reinhold, 1979; and (for Be2) V. E. Bondybey, Chem. Phys. Lett., 109,463 (1984).
3.2.1. Interaction of two identical AOs
The two MOs for the homonuclear diatomic molecule thus become
, -   1/Ž(xi + X2)    4>- =F^r-^ííží(;£l_Z2) (12)
[2(1
[2(1 - S)]1
So, starting off from two AOs, X\ and x2, we have obtained two MOs </>+ and </>_. This relationship between the number of AOs and the number of MOs they generate is a general one, even in more complex systems; n AOs give rise to n MOs. If we calculate the overlap between the MOs we get
1
<</> +  | </>-> = —      "„2<i7i «Zl I Xl> + Oil  I *2> - <X\ I *2> - <*2 I *2»
(1 + S — S — 1)
2(1 - S2)112 1
2(1 - S2)112 = 0
(13)
X2
Figure 3.1. Molecular orbital diagram showing the interaction between two identical atomic orbitals in a homonuclear molecule.
3.2.2. Interaction of two different AOs
*S 3-2h,M,olecula/ orbltal dla0ram showing the interaction between two different atomic orbitals in a heteronuclear molecule.
LoNMe
owe/
7-5  LINEAR COMBINATION OF ATOMIC ORBITALS
207
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -O.B -0.9 -1.0 -1.1 -1.2 -1.3
H atom energy \
\
\
HAA ~HAB.
»aa \
I"aa+»ab c
FIG. 7-6  Contributions to energy of H2 + at R — 2 in minimal basis LCAO-MO calculation.
208
7.   THE VARIATION METHOD
0.3 0.2 0.1 0.0 -0.1 ~ -0,2 <ü -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
FIG. 7-7  Separated atom energies and energies at an intermediate R for H2+. f *1
« (a.u.l
calcuEn7"*  E± + K for H2+. (__) Calculation described in text. (—) Exact