Physical Chemistry , 1st ed.

real (0)(1) (12.18)
It should be understood by now that there is not just one wavefunction for
a model system. There are a large number. Many times there are an infinite
number of wavefunctions, each with their own quantum numbers. Equation
12.18 can be rewritten to recognize the fact that the many wavefunctions are
all different and should be labeled. For example, using the label n(not to be
confused with the quantum number!):

n,real n(0)n(1) (12.19)

The entire group of wavefunctions for a model system is considered a complete
setof eigenfunctions. For a model system, the individual wavefunctions are
orthogonal; this fact will be important later. Such a situation is analogous to
the coordinates x,y, and zdefining three-dimensional space: the set (x,y,z)
represents a complete set of “functions” used to define any point in space. Any
point in 3-D space can be described as the appropriate combination of so
many xunit vectors, so many yunit vectors, and so many zunit vectors.*
The complete set of wavefunctions is similar. Such a set can be used to de-
fine the complete “space” of a system. The true wavefunction for a real, that is,
nonmodel, system can be written in terms of the complete set of ideal wave-
functions, just like any point in space can be written in terms ofx,y, and z.
Using first-order perturbation theory, any real wavefunction n,realcan be
written as an ideal wavefunction plusa sum of contributions of the complete
set of ideal wavefunctions m(0):

n,realn(0)

m

amm(0) (12.20)

where amis the coefficient multiplying each ideal m(0); they are called expan-
sion coefficients.Each real wavefunction n,realhas a different, unique set of ex-
pansion coefficients that define it in terms of the ideal eigenfunctions. A sum-
mation like equation 12.20 is called a linear combination,because it combines
the ideal wavefunctions, which are assumed to be raised to the first power
(which defines a linear type of relationship).
Although the process is lengthy, it is algebraically straightforward to deter-
mine what the expansion coefficients are for the correction to the nth real
wavefunction,n,real. Recall that each n,realis approximated initially by an
ideal n(0). The mthexpansion coefficient amfor the perturbation to the nth real
wavefunction n,realcan be defined in terms of the perturbation operator Hˆ,
the nth and mth idealwavefunctions m(0)and n(0), and the energies Enand
Emof the idealwavefunctions. Specifically,

am

(

E

m

(

n

(

0

0

)

)

)Hˆ





E



m

(0

n

(

)

0)d

 mn (12.21)

The restriction mncomes from the derivation of equation 12.21. The inte-
gration in the numerator is over the complete space of the system. The re-
quirement that this is nondegenerate perturbation theory also eliminates the
possibility that two energies En(0)and Em(0)might be equal due to degenerate
wavefunctions. (The extension of perturbation theory to degenerate wave-
functions will not be discussed here.)

12.6 Perturbation Theory 391

*The unit vectors in the x,y, and zdirections are labeled i,j, and k, respectively, so that
any point in 3-D space can be represented as xiyjzk.