NonCollinear Spin Density Functional: Ver. 1.0
Taisuke Ozaki, RCIS, JAIST
A two component spinor wave function is defined by
where
with a spatial function
and a spin function
.
In the notes we consider nonBloch functions, but the generalization
of the description to the Bloch function is straightforward.
Then, a density operator is given by
where should be a step function, but it is replaced
by the Fermi function in the implementation of OpenMX.
With the definition of density operator , a noncollinear
electron density in real space is given by
where
, and
is
a position eigenvector. The up and downspin densities
,
at each point are defined
by diagonalizing a matrix consisting of a noncollinear
electron densities as follows:
Based on the spinor wave function Eq. (1), the noncollinear
electron density Eq. (3), and the up and downspin densities,
the total energy noncollinear functional [1,2]
could be written by
where the first term is the kinetic energy, the second the electroncore
Coulomb energy, the third term the electronelectron Coulomb energy, and
the fourth term the exchangecorrelation energy, respectively.
Also the total electron density at each point is the sum of
up and downspin densities ,
.
Alternatively, the total energy can be expressed in terms of
the KohnSham eigenenergies
as follows:
where is a noncollinear exchangecorrelation potential
which will be discussed later on.
Considering an orthogonality relation among spinor wave functions,
let us introduce a functional :
The variation of with respect to the spatial wave function
is found as:
with
By setting the variation of with respect to the spatial wave
function to zero, and considering a unitary transformation
of
so that
can be diagonalized,
we can obtain the noncollinear KohnSham equation as follows:
We see that the offdiagonal potentials produce explicitly a direct interaction
between and spin components in this  coupled
equation. The offdiagonal potentials consist of the exchangecorrelation
potential and the other contributions such as
spinorbit interactions.
The matrix in Eq. (4) which relates the noncollinear electron densities
to the up and downspin densities is expressed by a rotation
operator [4]:
with Pauli matrices
where is a unit vector along certain direction, and
a rotational angle around .
Then, consider the following twostep rotation of a unit vector
(1,0) along the zaxis:
 First, rotate on the yaxis
 Second, rotate on the zaxis
The unit vector (1,0) along the zaxis is then transformed as follows:
where
Thus, if the direction of the spin is specified by the Euler
angle (), the matrix in Eq. (4) is given by
the conjugate transposed matrix of Eq. (15).
The meaning of Eq. (4) becomes more clear when it is written
in a matrix form as follows:
We see that the Umatrix diagonalizes the total (average) noncollinear
spin matrix rather than the noncollinear spin matrix of each state .
Since the exchangecorrelation term is approximated by the LDA or GGA,
once the noncollinear spin matrix is diagonalized, the diagonal up and
downdensities are used to evaluate the exchangecorrelation potentials
within LDA or GGA:
Then, the potential
is transformed to the noncollinear
exchangecorrelation potential as follows:
where
The Euler angle () and the up and downspin densities
(,
) are determined
from the noncollinear electron densities so that the following
relation can be satisfied:
After some algebra, they are given by
Then, it is noted that the effective potential in Eq. (11)
can be written in Pauli matrices as follows:
where
As well, the noncollinear spin density can be also written in Pauli matrices
as follows:
with
where is a unit matrix.
In OpenMX, the spinorbit coupling is incorporated through
jdependent pseudo potentials [3].
Under a spherical potential,
a couple of Dirac equations for the radial part is given by



(36) 



(37) 
where and are the majority and minority components of
the radial wave function. (1/137.036 in a.u.).
and for
and
, respectively.
Combining both Eqs. and eliminating , we have
the following equation for :



(38) 
with
By solving numerically Eq. (38) and generating jdependent
pseudo potential
by the Troullier and Martine
(TM) scheme, we can define a general pseudopotential by
where
for
and
and for
and
The and
are constituents of
the eigenfunction of Dirac equation.
Since
and
, the degeneracies
of and are and , respectively.
In the use of the pseudopotential defined by Eq. (40),
it is transformed to a separable form.
By introducing a local potential
which approaches
as increases,
the jdependent pseudo potential is divided into two contributions:
The nonlocal potentials
and
are nonzero within a certain radius.
Then, the pseudopotential defined by Eq. (40) is written by
The nonlocal part is transformed by the Blochl projector
into a separable form:
with
where
and is an orthonormal set defined by
a norm
,
and is calculated by the following GramSchmidt orthogonalization:
Similarly,
with
Moreover, by unitary transforming the complex spherical harmonics functions
into the real spherical harmonics function ,
we obtain the following expressions:



(59) 



(60) 



(61) 



(62) 



(63) 



(64) 



(65) 



(66) 
with
where the real spherical harmonics functions are
denoted by ,
, and
for , , and orbitals, respectively.
In conjunction with the onsite exchange term of the unrestricted
HartreeFock theory, the total energy of a noncollinear LDA+U method
could be defined by
with
where is a site index, an angular momentum quantum number,
a multiplicity number of radial basis functions, and an
organized index of . is an diagonalized
occupation matrix with the size of
.
The is the effective Coulomb electronelectron interaction energy.
Also, is given by Eq. (5).
It should be noted that the occupation matrix is twice as size as
the collinear case. In this definition it is assumed that the exchange
interaction arises when an electron is occupied with a certain spin
direction in each localized orbital.
Considering the rotational invariance of total energy with respect
to each subshell , Eq. (100) can be transformed as follows:
where
.
In this Eq. (101), 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.[5], 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 addition, the functional is rotationally invariant in the spinspace.
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 total energy can be expressed in terms of the
KohnSham eigenenergies
as follows:
where
and
are the double counting
corrections of LDA and Uenergies, respectively.
The occupation number (which is written by an italic font, while
the electron density, appears in Sec. 1, is denoted by a roman font)
is defined by
where, to count the occupation number , we define three occupation number
operators given by
onsite
full
dual
where
is the dual orbital of a original
nonorthogonal basis orbital
, and is defined by
with the overlap matrix between nonorthogonal basis orbitals.
Then, the following biorthogonal relation is verified:
The onsite and full occupation number operators have been
proposed by Eschrig et al. [6]
and Pickett et al. [7], 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:
where is the density matrix defined by
with a density operator:
For three definition of occupation number operators,
onsite, full, and dual,
the occupation numbers are given by
onsite
full
dual
The derivative of the total energy Eq. (99) with respect to LCAO coefficient
is given by
with
onsite
full
dual
Substituting Eqs. (116)(118) for the second term of Eq. (115),
we see
onsite
full
dual
Therefore, the effective projector potentials
can be expressed by
onsite
full
dual
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
Eqs. (123) and (124) because of the overlap.
The force on atom is evaluated by
The first term can be calculated in the same way as in the collinear case.
The second term is evaluated as follows:
Considering
and
,
the first and second terms in Eq. (127) can be transformed into
derivatives of the overlap matrix. The third term in Eq. (127) means
that only the differentiation for the overlap matrix is considered,
And it 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:



(128) 



(129) 



















(130) 



(131) 



(132) 
After diagonalizing each subshell matrix consisting of occupation numbers,
we introduce a polarized redistribution scheme given by Eq. (130) while keeping
Eq. (129). Then, by a back transformation Eq. (130), 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,...
Define
Then, the density of states, , is given by
Also, the Mulliken populations, , are given by
The local spin direction is determined by
The contribution to the total energy arising the Zeeman term is given by
where
The vector components of the spin magnetic moment are given by
with
where
is given by
After some alegebra we have
The vector components of the orbital magnetic moment are given by
where
.
Noting that



(152) 



(153) 



(154) 



(155) 



(156) 
and considering a unitary transformation of the spherical harmonic functions
into a set of real harmonic functions defined by
it can be shown that
It is noted that the expectation values of in terms of the real
harmonic functions are purely imaginary numbers.
The unitary transformation for the other components can be found
in a subroutine 'Set_Comp2Real()' in 'SetPara_DFT.c'. Thus, one can obtain
the matrix representation for in terms of the real harmonic functions.

 1

U. Von Barth and L. Hedin, J. Phys. C: Solid State Phys. 5, 1629 (1972).
 2

J. Kubler, KH. Hock, J. Sticht, and A. R. Williams,
J. Phys. F: Met. Phys. 18, 469 (1988).
 3

G. Theurich and N. A. Hill, Phys. Rev. B 64, 073106 (2001).
 4

J. J. Sakurai, Modern Quantum Mechanics.
 5

S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and
A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
 6

H. Eschrig, K. Koepernik, and I. Chaplygin, J. Solid State Chem. 176, 482 (2003).
 7

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