LDA+U Method: Ver. 1.0
Taisuke Ozaki, RCIS, JAIST
In conjunction with onsite terms of the unrestricted HartreeFock
theory, the total energy of a LDA+U method [1] within the collinear
spin treatment could be defined by



(1) 
with
where is a site index, an angular momemtum quantum number,
a multiplicity number of radial basis functions, a spin
index, and an organized index of . is an diagonalized
occupation matrix. is the effective Coulomb electronelectron
interaction energy.
Considering the rotational invariance of total energy with respect
to each subshell , Eq. (2) can be transformed as follows:
In the Eq. (3), although offdiagonal occupation terms in each subshell
are taken into account, however, those between subshells are neglected.
This treatment is consistent with their rotational invariant functional
by Dudarev et al. [2], and is a simple extension
of the rotational invariant functional for the case that a different
Uvalue is given for each basis orbital indexed with .
In this simple extension, we can not only include multiple dorbitals
as basis set, but also can easily derive the force on atoms in a simple
form as discussed later on.
The can be expressed in terms of the KohnSham eigenenergies
as follows:
where
and
are the double couting
corrections of LDA and Uenergies, respectively.
The occupation number may be defined by
where, to count the occupation number , we define three occupation number
operators given by
onsite



(6) 
full



(7) 
dual



(8) 
where
is the dual orbital of a original
nonorthogonal basis orbital
, and is defined by



(9) 
with the overlap matrix between nonorthogonal basis orbitals.
Then, the following biorthogonal relation is verified:



(10) 
The onsite and full occupation number operators have been
proposed by Eschrig et al. [3]
and Pickett et al. [4], respectively.
It is noted that these definitions do not satisfy a sum rule that the
trace of the occupation number matrix is equivalent to
the total number of electrons, while only the dual occupation number
operator fulfills the sum rule as follows:



(11) 
where is the density matrix defined by
with a density operator:



(13) 
The notes limit the discussion to nonBloch wave functions for simplicity,
but the extension is straightforward.
For three definition of occupation number operators, onsite, full, and dual,
the occupation numbers are given by
onsite



(14) 
full



(15) 
dual



(16) 
The derivative of the total energy Eq. (1) with respect to LCAO coefficient
is given by
with
onsite



(18) 
full



(19) 
dual



(20) 
Substituting Eqs. (18)(20) for the second term of Eq. (17), we see
onsite



(21) 
full



(22) 
dual



(23) 
Therefore, the effective projector potentials
can be expressed by
onsite



(24) 
full



(25) 
dual



(26) 
It is clear that the effective potentials of onsite and full
are Hermitian. Also, it is verified that the effective potential
of dual is Hermitian as follows:
It should be noted that in the full and dual the of the
site can affect the different sites by the projector potentials
by Eqs. (25) and (26) because of the overlap.
The derivative of the total energy with respect to atomic coordinates
consists of two contributions:
The first term can be evaluated in the same way as in the LDA.
The second term is given by
Considering
and
,
the first and second terms in Eq. (30) can be transformed into
derivatives of the overlap matrix. The third term in Eq. (30) is
analytically differentiated, since it contains just twocenter integrals.
The LDA+U functional can possess multiple stationary points
due to the degree of freedom in the configuration space
of occupation ratio for degenerate orbitals. If electrons are
occupied with a nearly same occupancy ratio in degenerate
orbitals at the first stage of SCF steps,
the final electronic state often converges a stationary
minimum with nonorbital polarization after the SCF iteration.
Also, it is often likely that electrons are disproportionately
occupied in some of degenerate orbitals due to the exchange
interaction, which is socalled 'orbital polarization'.
As an example of the multiple minima,
we can point out a cobalt oxide (CoO) bulk in which dorbitals
of the cobalt atom are split to and states, and
the five of seven delectrons are occupied in and
states of the majority spin, and remaining two delectrons are
occupied in the state of the minority spin.
Then, it depends on the initial occupancy ratios for the
states of the minority spin how the remaining two delectrons
are occupied in three states.
If the initial occupancy ratios are uniform, we may arrive at
the nonorbital polarized state. In fact, unless any special
treatment is considered for the initial occupancy ratios,
we see the nonorbital polarized state of the CoO bulk.
In order to explore the degree of freedom for the orbital
occupation, therefore, it is needed to develop a general method
which explicitly induces the orbital polarization.
To induce the orbital polarization, a polarized redistribution scheme
is proposed as follows:
After diagonalizing each subshell matrix consisting of occupation numbers,
we introduce a polarized redistribution scheme given by Eq. (33) while keeping
Eq. (34). Then, by a back transformation Eq. (35), we can obtain
a polarized occupation matrix for each subshell. This polarized redistribution
scheme is applied during the first few SCF steps, and then no modification
is made during subsequent SCF steps.
This proposed scheme maybe applicable to a general case:
any crystal field, any number of electrons
in the subshell, and any orbitals: p,d,f,...
In the orbital optimization within LDA+U, let us assume that the effective
Upotential in the LDA+U method is applied
to the primitive basis orbital instead of the optimized basis orbital ,
which is more natural in a physical sense than the opposite assumption.
A KohnSham (KS) orbital in the orbital optimization method
is expressed by a linear combination of primitive orbitals :
where
,
, and are LCAO
coefficients for contracted and primitive orbitals, respectively, and
contraction coefficients.
For simplicity we consider an nonBloch expression of the oneparticle
wave functions, but the extention of the below description to Bloch wave
functions is straightforward.
Assuming that the occupation number operators defined by Eqs. (6)(8)
are constructed by the primitive orbitals, we have the occupation numbers
for the onsite, full, and dual given by
onsite



(37) 
full



(38) 
dual



(39) 
where
is the primitive density matrix defined by
with a primitive density operator:
Moreover, by defining a contracted density operator:
we have the contracted density matrix given by
Then, the primitive density matrix is written by the contracted
density matrix as follows:
Considering the variation of the total energy Eq. (1) with respect to ,
we find the effective potentials of the LDA+U method with respect to
the primitive basis orbital. They are given by the same expression
as Eqs. (24)(26), while the occupation number is given by Eqs. (37)(39).
After the Hamiltonian matrix with respect to the primitive basis orbital
is constructed, it is transformed to that of the optimized basis orbital
as follows:
The Hamiltonian matrix with respect to the contracted basis orbital
is diagonalized.
The procedure is summarized as follows:
 diagonalize the contracted Hamiltonian
 calculate the contracted density matrix by Eq. (43)
 calculate the primitive density matrix by Eq. (44)
 calculate the occupation number by Eq. (37), (38), or (39)
 construct the Hamitonian by Eq. (24), (25), or (26)
 contract the Hamitonian by Eq. (45)
 return 1
Although the optimization procedure of the contracted coefficients
is not discussed here, it can be easily verified that the same procedure
as in the LDA method is derived. Thus, the orbital optimization can be
performed within the LDA+U method as well as the LDA method.

 1

M. J. Han, T. Ozaki, and J. Yu,
Phys. Rev. B 73, 045110 (2006).
 2

S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and
A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
 3

H. Eschrig, K. Koepernik, and I. Chaplygin,
J. Solid State Chem. 176, 482 (2003).
 4

W. E. Pickett, SC. Erwin, E. C. Ethridge,
Phy. Rev. B 58, 1201 (1998).
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