Theory of MX Chain Compounds - Material Designs and New Physical Properties in MX-and MMX-Chain Compounds - page 114

8.2.1 Ground State

We start our analysis of the extended Peierls-Hubbard hamiltonian in Eq. ( 8.2 ),
based on a Hartree-Fock HF approximation. Such a treatment is elementary, but it is
a good exercise to prepare ourselves before advanced calculations. In this approxi-
mation, we simply replace the interaction terms in Eq. ( 8.2 ) using this recipe:

n l" n l# ! n l" hn l# iþhn l" in l# hn l" ihn l# i
n l n lþ 1 ! n l hn lþ 1 iþhn l in lþ 1 hn l ihn lþ 1 i

n o

;

X

C lþ 1 s C ls hC ls C lþ 1 s iþ : hC lþ 1 s C ls ihC ls C lþ 1 s i

h.c (8.4)

s

h...i

where the brackets are the mean fields to be determined self-consistently. In

practical calculations, we start from

hC ls C lþ 1 s i¼m ls with n ls being
and m ls appropriate c -number distributions. Such an assumption enables us to make
a one-body mean-field hamiltonian as

hn ls i¼n ls and

X X

N

1 ððt 0 þ V m ls ÞC lþ 1 s C ls þ

2

H ePH2 ; HF ¼ h.c : V m ls j j Þ

s¼";# l¼

þ U X

l¼ 1 ðn l" n l# þ n l" n l# n l" n l# ÞþV X

N N

l¼ 1 ðn l n lþ 1 þ n l n lþ 1 n l n lþ 1 Þ

S X

2 X

N
l¼ 1 ðq lþ 1 q l Þn l þ

N

S

q l :

ð 8 : 5 Þ

l¼ 1

Assuming a classical distribution for q l , we can now solve this hamiltonian very
easily. Since we only treat the case of zero temperature throughout this article, the
expectation values like

2 one-electron
orbitals for each spin ( N e is the total electron number). We then try iterations until
self-consistency conditions as

hn ls i are obtained by selecting the lowest N e =

hC ls C lþ 1 s i¼m ls are satisfied. Thus,
this is a type of mean-field treatment, while it is educational to emphasize that this is
a variational method at the same time. The variables like

hn ls i¼n ls and

n ls there work as
variational variables that minimize the expectation value of the original hamilto-
nian through the Slater determinant determined by Eq. ( 8.5 ). This variational
aspect, which is easily proved by taking functional derivatives of each one-electron
orbital function, is very important, since modern methods like a density-matrix
renormalization group DMRG treatment are also variational methods in a much
more extended meaning.
Returning to the CDW problem, it is natural to set q l ¼ q 0 ð

l ,

1 Þ n ls ¼ 1 = 2

l ,and

m ls ¼ m . Moreover, using the Hellmann-Feynman theorem,

þdnð 1 Þ