Concise Physical Chemistry

c20 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

320 QUANTUM MOLECULAR MODELING

There are many possible values ofR, each of which leads to a unique value of the

electronic energyEelectronic. AlthoughRis constant for any single calculation, the

total energy of the system is a function ofR:

Etotal=Eelectronic+

1

R

This means that a curve ofEtotalvs.Rcan be drawn. From this point on, we shall

drop the subscripts onE, taking the nature ofE, electronic or total, to be clear from

context.

By theLCAOapproximation, a molecular orbital can be expressed as aLinear

Combination of two hydrogenicAtomicOrbitals:

ψ=N 1 e−rA±N 2 e−rB

The atomic orbitals arebasis functionswhich define a vector space that includes the

molecular orbital. The LCAO basis set is not complete, so the molecular orbital we

obtain will not be correct. If functionsN 1 e−rAandN 2 e−rBare normalized hydrogen

1 sorbitals, we call them 1sAand 1sBand the approximate molecular orbital for H+ 2

isψH+ 2 = 1 sA± 1 sB. Physically, one basis function represents hydrogen atom A at

some distance from proton B, while the other basis function represents hydrogen

atom B at some distance from proton A. Neither basis function alone recognizes the

simultaneous interaction of the electron with both nuclei. It is this interaction we seek

because it brings about the chemical bond.

The energy of the positive combinationψH+ 2 = 1 sA+ 1 sBis

E=

∫

ψHˆψdτ=N^2

∫

( 1 sA+ 1 sB)

[

−^12 ∇^2 −

1

rA

−

1

rB

]

( 1 sA+ 1 sB)dτ

whereN^2 is the product of the two normalization constants and the integration is

taken over the entire spaceτ. If we expand and simplify according to the symmetry

of the problem, the energy gives three functions often denotedJ,K, andS:

J=

(

1 +

1

R

)

e−^2 R

K=

(

S

R

− 1 +R

)

e−^2 R

and

S=

(

1 +R+

R^2

3

)

e−R