3.3 Crystal Orbital (CO) Calculations
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3.3.2 Wavefunction of 1D Polymer
The wavefunction ψ(x) describing 1D polymer can be translated by the translation
length a and the translated one, ψ(x + a), is “essentially the same” with the original
wavefunction except for the phase factor λ being a complex number having the
absolute value of unity. This relationship is represented by
ψ(x + a) = λψ(x)
(3.31)
Then for the successive translation one obtains
ψ(x + 2a) = λ
2
ψ(x)
ψ(x + 3a) = λ
3
ψ(x)
ψ(x + 4a) = λ
4
ψ(x)
. . .
ψ(x + Na) = λ
N
ψ(x) = ψ(x)
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(3.32)
The last equation in the above comes from the very periodic boundary condition
shown in Fig. 3.10, which implies
λ
N
= 1
(3.33)
This equation is the usual circle equation giving λ as follows:
λ = exp
2π iq
N
(q = 0, 1, 2, . . . .., N − 1)
(3.34)
where N is formally infinite and i imaginary unit. By setting
2π q
Na
≡ k
(3.35)
one can write λ more simply as follows:
λ = exp[ika]
(3.36)
Hence, Eq. (3.31) is changed into
ψ(x + a) = exp[ika]ψ(x)
(3.37)
where k is the reciprocal translation vector, often called wave vector or wavenumber,
having the reverse dimension of the translation length in the range