Microscopic Modelling of Correlated Low-dimensional Systems

Chapter 2: Method 23
(P α
ll − Sα ll − Sα α
li (Pii − S α ii) −1 S α il )(N α l )−1ul = 0 (2.40)
when linearizing and solving this equation, the downfolded bands are obtained. This method
has proved to be extremely successful for systems such as high-Tc cuprates [110] or low-
dimensional inorganic quantum spin systems [111].
2.2.4 NMTO
The order-N muffin tin orbital (NMTO) (also known as third-generation MTOs) method
provides a formalism more consistent than MTO and LMTO, in which the interstitial region
is treated more accurately. The NMTO method treats the sphere and the interstitial equally
by working with MTO-type functions ψL(En, r − R) localized around site R and calculated
at fixed energies En both inside the sphere and in the interstitial (assumed to have a flat
muffin tin potential). The NMTO basis function is then defined to be a linear combination
of N, such functions evaluated at N energies [6], [8]
χ
NMT O
R,L
(ɛ, r) =
N� �
n=0 R ′ L ′
ψL ′(En, r − R ′ )L N nL ′ R ′ ,LR (ɛ, r) (2.41)
where L N n (ɛ) is the transformation matrix that includes the idea of screening (mixing states
on different sites) and a linear combination of states evaluated at N fixed energies. The
basic idea is then, instead of using one MTO with one fixed energy (N = 1) ɛ, N-energies
(N > 1) ɛ0, ..., ɛN are used to construct the MTO, which then have errors proportional to
(ɛi − ɛ0), ..., (ɛi − ɛN). This is analogous to using Lagrange or Newton interpolation and is
far more practical.
Other important characteristics of these approach:
• The NMTO’s are uniquely suited in materials with many bands and it provides a
quantitative description of a one-electron mean-field theory with a minimal basis.
• The NMTO basis is a minimal set spanning all states in a wide energy range
• The energy selective and localized nature of NMTO basis makes the NMTO set flexible
and may be chosen as truly minimal (=span selected bands with as few basis functions
as there are bands.)