Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

MANY-ELECTRON ATOMIC SYSTEMS 263

(a)

• 
  
  
  
  • 
    
    
    
    • s
    
    
    
    
  
  
  
  

px

(b)

+

pz

s

+

FIGURE 16.11 Favorablespxand unfavorablespzoverlap of orbitals depending upon orbital

symmetry. Diagram a depicts bond formation but orbital overlap exactly cancels in diagram b.

are combined as basis functions (vectors) to describe a chemical bond, the result

may be favorable, as in Fig. 16.11a, resulting in enhancement of the probability

density between bonded atoms. The overlap of anspositive orbital and apxpositive

orbital is positive. Or it may be unfavorable, depending on the orbital symmetry, as

in Fig. 16.11b where any++overlap between one lobe of the 2pzorbital with the 1s

orbital is canceled by the negative+−overlap of the other. In analyzing Fig. 16.11,

we are asking the question of whetherporbitals can or cannot be used as basis vectors

to describe a favorable overlap, thus a chemical bond. The answer isyesfor thepx

orbital and no for thepz. A natural question is: What is so special about thepx

orbital? Couldn’t we just rotate thepzorbital a quarter turn into a favorable overlap?

The answer is that there isnothingspecial about thepxorbital and yes, we can

rotate thepzorbital into a position of favorable overlap, but in so doing, we rotate

thepxorbitaloutof its position of favorable overlap so we haven’t really changed

anything. We are merely saying that if one of theporbitals is favorable, the other two

orthogonal orbitals are not. The orbitals don’t care how we label them.

16.10 MANY-ELECTRON ATOMIC SYSTEMS

If we regard each electron as obeying an orbital (function) that is independent of

all other electrons except that it moves in the potential field of the nuclei and their

averageelectrostatic forces, we get a wave function for the entire system that is

approximated by a product of many one-electron orbitals called theHartree product:

ψHartree=ψ 1 (r 1 )ψ 2 (r 2 ),..., ψN(rN)=

∏N

i= 1

ψi(ri)

This product produces N integrodifferential equations:

hˆiψi(ri)=εiψi(ri), i= 1 , 2 ,...,N

where the operatorhˆiincludes the kinetic energy and the potential energy of attraction

along with the potential energy of interelectronic repulsionVi:

hˆi=

[

−^12 ∇i^2 −

Z

ri

]

+Vi(ψi(rj)), j =i