Orbital Optimization Method: Ver. 1.0
The original has been published
in Phys. Rev. B 67, 155108 (2003)
Taisuke Ozaki, RCIS, JAIST
Abstract:
A simple and practical method for variationally optimizing numerical
atomic orbitals used in density functional calculations is presented
based on the force theorem. The derived equation provides the same
procedure for the optimization of atomic orbitals as that for
the geometry optimization.
The optimized orbitals well reproduce convergent results calculated by
a larger number of unoptimized orbitals.
In addition, we demonstrate that the optimized orbitals significantly
reduce the computational effort in the geometry optimization,
while keeping a high degree of accuracy.
I. INTRODUCTION
During the last decade, to extend the applicability of density functional
theories (DFT) to realistic large systems, great efforts have been done for
developing O() methods of the eigenvalue problem [1,2,3,4,5,6]
and for making efficient and accurate localized orbitals
[7,8,9,10] as a basis set being suitable for O() methods.
Among these studies, one of important and unresolved problems is
how atomic orbitals as a basis set are constructed to maximize both
the computational efficiency and accuracy.
One expects that a basis set such as double valence orbitals
with polarized orbitals for valence electrons provides
a way for balancing a relatively small computational effort and
a considerable degree of accuracy. Along this line, accurate basis sets
were constructed in several ways [7,8,9,10].
Kenny et al. constructed a basis set, so that atomic orbitals span the subspace
defined by selected and occupied states of reference systems as much as possible
[7].
Junquera et al. optimized the shape and cutoff radii of atomic orbitals
for reference systems by using the downhill simplex method [8].
However, the transferability of these optimized orbitals might be restricted
to systems similar to the reference systems used for the optimization
in terms of atomic environments and states such as the coordination number
and the charge state.
A more complete treatment is to optimize atomic orbitals of each atom
located on different environments in a given system [10].
In addition, the complete optimization procedure should be simple and
efficient practically. In this paper, to overcome the difficulty,
we present a simple and practical method, based on the force theorem, for
variationally optimizing numerical atomic orbitals of each atom
in a given system.
II. VARIATIONAL OPTIMIZATION
Let us expand a KohnSham (KS) orbital of a given system
using numerical atomic orbitals
in a form of
linear combination of atomic orbitals (LCAO):
where is a site index,
an organized orbital index,
and
.
For simplicity we consider only nonspinpolarized systems and an nonBloch
expression of the oneparticle wave functions, but the extentions
of the below description to spinpolarized systems and Bloch wave functions
are straightforward.
Note that a radial wave function depends on not only an angular
momentum quantum number , but also a site index , a multiplicity
index , and a magnetic quantum number .
To give a variational degree of freedom of
,
we furthermore expand
using primitive orbitals
as follows:
where
, in which the indices and denote the same
as those of the index , and
.
Note that a primitive radial wave function , which is discussed
later on, is independent on , and that the coefficients
are independent variables on the eigenstate .
Substituting Eq. (2) into Eq. (1), we have
Although the expansion of a KS orbital by Eq. (3) is linearized
for each variable
or , however, the
primitive orbitals are expanded by the product of two
variables
and in a nonlinearized form.
Therefore, it is difficult to directly find the minimum of the KS total
energy for the ground state with respect to
and as a generalized eigenvalue problem which can be
derived in the usual LCAO.
To avoid the difficulty, here, we propose a two step optimization scheme,
in which the coefficients are optimized after
are determined with a set of fixed .
Considering
for the KS total energy with the orthonormalization relation
among
oneparticle wave functions and fixed contraction
coefficients , we have a wellknown KS matrix equation
with respect to the coefficient
as follows:
where is a KS Hamiltonian, and
is a KS eigenvalue
of the system. There is no restriction to solve Eq. (4) or to find
. So, we can use
any solution method which could be an exact diagonalization method,
iterative methods such as CarParinello (CP) method [11]
and conjugate gradient (CG) method [12], and O() methods.
On the other hand, regarding
as dependent variables
on and assuming that the KohnSham equation is solved
selfconsistently with respect to
, we can derive the
following equation based on the force derivation of nonorthogonal orbitals
as follows:
where is an occupancy number for the eigenstate ,
a bondorder, and
an energy bondorder.
The final equation in Eq. (5) is derived by taking into account Eq. (4)
and the orthonormalization relation
.
It should be noted that Eq. (5) excludes any derivative,
and that
and
only
have to be evaluated for the contracted atomic orbital
in Eq. (5), which implies that additional computational costs are not
required to evaluate Eq. (5).
Once we obtain a selfconsistent solution of the KohnSham equation,
Eq. (4), with a set of given coefficients ,
then Eq. (5) gives the gradient of with respect to
within small computational costs.
This fact shows apparently that
the atomic orbitals can be optimized variationally in the same two
step procedure as that of the geometry optimization in terms of
instead of atomic positions. Therefore, the contraction
coefficients are optimized iteratively, coupled with
the selfconsistent solution of Eq. (4), as follows:
where an optimum is determined, so that the norm
of the gradients at ,
becomes a minimum with respect to under the fixed
and
.
Substituting Eqs. (5) and (6) into Eq. (7), and considering
with the fixed
and
,
we have for the minimum of
with
where
with
and
.
Taking into account the sparseness of both the Hamitonian and overlap matrices
in the real space, we find that the computational efforts to evaluate Eqs. (8)
and (9) scale linearly. Therefore, we can easily evaluate , leading no
intensive computational demands.
After achieving the selfconsistent field (SCF) for Eq. (4) with respect
to the coefficients
by an usual SCF procedure,
the contraction coefficients are updated by Eq. (6) and
renormalized so that
is normalized.
Thus, the two step optimization scheme enables us to optimize the
contraction coefficients along the stationary minimum
line of the KS total energy functional with respect to
,
while keeping
.
It is found that about five iterative procedures of the two step optimization,
which includes the solution of Eq. (4) and the optimization of
by Eq. (6), are enough to accomplish a sufficient convergence
of for our test systems.
Again it should be mentioned that the bondorder and the energy bondorder
are required for only the contracted atomic orbital
in Eq. (5). This is a crucial point to make our optimization procedure
efficient, since the KohnSham equation based on Eq. (4) can be solved
for not large , but small
.
Once the contraction coefficients are fixed
after the orbital optimization, the Hamiltonian and overlap matrices
are directly constructed for the small
without
constructing the elements for the larger primitive orbitals,
since we can directly utilize the contracted orbital
as a numerical table because of the use of numerical orbitals,
which is also a reason why the optimization scheme could be totally
efficient.
A rather technical but important problem still remains in the application
of the two step optimization method. To avoid the orbital optimization
to a local minimum, we have developed the following careful procedure
to provide a good initial guess for a set of coefficients :
(I) The partition of the system.
A cluster is constructed including the nearest neighboring atoms
for each atom .
(II) Solving of the KohnSham equation of each cluster.
By non selfconsistently solving the KohnSham equation of each cluster
which is centered on an atom , we obtain the
coefficients for the atom of an oneparticle
wave function
with an eigenvalue
.
(III) Construction of from .
Then, is given as
where is the Fermi function,
a local chemical potential for the cluster , and N
a normalization factor. For , the coefficients are generated
by the GramSchmidt orthonormalization from and
in order of the magnitude of
.
To find a good initial guess for the coefficients ,
we tried to estimate the ratio of coefficients of
the eigenstates of the whole system expanded by from
the local cluster for each atom by the above treatment. Then, the
eigenstates of the cluster are weighted by the Fermi function, so that
the contribution of the lower states is taken into account
as much as possible. Also, the contraction coefficients
for are generated using the GramSchmidt method to avoid the
overcompleteness of contracted basis orbitals.
We found that the procedure provides good initial coefficients
in all of our test systems, and did not observe
that the orbital optimization is trapped to any serious local
minimum, while the other trial was trapped to a local minimum often.
The additional cost for the above procedure (I)(III) is almost
negligible, when the orbital optimization method is applied
as a preconditioning of the geometry optimization as discussed later on.
III. PRIMITIVE ORBITALS
The primitive orbitals we used are eigenstates of an atomic
KohnSham equation with confinement pseudo potentials [8,13].
To vanish the radial wave function of the outside of the confinement
radius , we modify the atomic core potential
in the all electron calculation of an atom and the generation of pseudo
potential as follows:
where , , , and are determined, so that the value and
the first derivative are continuous at both and .
Figure 1 shows radial wave functions for of a carbon atom under
the confinement pseudo potential. The eigenstates construct an orthonormal
basis set at the same atomic position and vanish beyond
within the double precision. Because of the complete vanishing tail of
numerical orbitals, we find that nonzero elements of Hamiltonian and overlap
matrices can be exactly proportional to the number of atoms.
In Fig. 2 the total energy for a carbon dimer calculated using the
eigenstates as a basis set is shown as a function of
the number of orbitals for various cutoff radii .
Factorized norm conserving pseudo potentials [13] and the local
density approximation (LDA) [14] to the exchangecorrelation interactions
were used in our all DFT calculations.
Also the real space grid techniques were used with the energy cutoff
of 113 (Ryd) for numerical integrations [8].
As the cutoff radius and the number of orbitals increase, the total energy
converges systematically. Thus, we see that the primitive orbitals
itself are systematic basis sets controlled by two simple
parameters, the cutoff radius and the number of orbitals, in the same manner
as spherical wave basis sets [9].
In addition, a relatively small number of orbitals may be needed to obtain
the convergent result compared to the spherical wave basis sets,
since the primitive basis set is prepared for each element,
unlike the spherical wave basis sets [9].
These are reasons why we use the eigenstates of
an atomic KohnSham equation with the confinement pseudo potentials
as the primitive orbitals.
A systematic study for convergence properties as a function of
the cutoff radius and the number of orbitals will be presented
for several elements including first row elements, alkaline metals
and transition metals elsewhere.
For the later discussion, here, we introduce an abbreviation of
the basis orbital as C4.5s62p62, where C indicates the
atomic symbol, 4.5 is the cutoff radius (a.u.) used
in the generation, s62 means that two optimized orbitals are
constructed from six primitive orbitals for the sorbital,
and the symbol signifies the restricted optimization that
the radial wave function is independent on the index .
In case of snn such as s66,
corresponding to no optimization, snn can be simplified as sn.
IV. NUMERICAL RESULTS
Figure 3 shows the convergence properties of total energies for
a carbon dimer C, a methane molecule CH, and the diamond
as a function of the number of unoptimized and optimized orbitals.
The orbital optimization was done by five iterative steps according
to Eq. (6), in which each step includes ten SCF loops.
We see that the unoptimized orbitals provide systematic and rapid
convergent results for not only molecules C and CH,
but also a bulk system diamond, as the number of orbitals increase.
Moreover, remarkable convergent results are obtained using the optimized
orbitals for all systems. The small optimized orbitals rapidly converge
to the total energies calculated by a larger number of unoptimized
orbitals, which implies that the computational effort can be reduced
significantly with a high degree of accuracy. For three systems the
effect of the restriction for the orbital optimization is almost negligible,
which encourages us to use the restriction, since the restricted optimization
guarantees the rotational invariance of the total energy.
In Fig. 4 the radial parts of the minimal orbitals obtained by the restricted
optimization for the diamond are shown with those of the lowest primitive
orbitals of a carbon atom for comparison. It is observed that the tails of both
the optimized s and porbitals shrink compared to the primitive orbitals,
which clearly reveals that the basis orbital can automatically vary within
the cutoff radius to minimize the total energy.
Finally, as an illustration of the orbital optimization, we performed
the geometry optimization with the orbital optimization as a preconditioning
for the most stable conformer of a neutral glycine
molecule [15,16] which is the smallest amino acid.
Before doing the geometry optimization, the orbital optimization was
performed by five iterative steps, which includes ten SCF loops per
step, for an initial structure optimized by a molecular mechanics (MM).
Then, the geometry optimization was done using the optimized orbitals
by fifty steepest decent (SD) steps with a variable prefactor for accelerating
the convergence, which includes twenty SCF loops per step.
The optimized geometrical parameters are given in Table I together
with the total energy and the computational time per MD step.
In case of the unoptimized orbitals SN, TN, and TNDP, as the number
of orbitals increase, we find the decrease of the total
energy and the convergent geometrical parameters comparable
to the experimental [16] and the other theoretical values [15].
Although there are some deviations in the optimized parameters calculated
using TNDP from the other theoretical values [15],
the deviations may be attributed to the pseudo potentials rather than
the basis orbitals, since we verified that the optimized parameters of
the glycine depend on the cutoff radii in the pseudo potential generation.
Comparing to the unoptimized and optimized minimal orbitals SN and SN,
it is found that the geometrical parameters are significantly improved
without increasing the computational time. In case of the optimized orbitals
SNP remarkable improvements are obtained in both the geometrical parameters
and the computational time. The optimized orbitals SNP provide a convergent
result comparable to TNDP with a great reduction of the computational time.
The computational time required for the orbital optimization of SN occupies
only 3 % of that of the whole calculation. So the orbital optimization can be
regarded as a preconditioning before doing the geometry optimization or
the molecular dynamics. Of course, it is possible to perform the orbital
optimization during the geometry optimization.
It is worth mentioning that the orbital optimization can be combined with
an O() method [1,2,3,4,5,6],
since only
and
, which are calculated by the O() method,
are required in Eq. (5). Therefore, the orbital optimization can
be applied to largescale systems in O() operations.
V. CONCLUSIONS
To conclude, we have developed a simple and practical method based on
the force theorem for variationally optimizing numerical atomic orbitals
used in density functional calculations. The optimization algorithm similar
to the geometry optimization allows us to fully optimize atomic orbitals
within a cutoff radius for each atom in a given system.
The illustration of geometry optimization with the orbital optimization
for a small molecule clearly shows that the small optimized orbitals
promise to greatly reduce the computational effort with a high degree
of accuracy.
ACKNOWLEDGMENTS
We would like to thank H. Kino for helpful suggestions and discussions.
This work was partly supported by NEDO under the Nanotechnology Materials
Program.

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Table 1:
Optimized geometrical parameters (Å and degrees) of
the most stable conformer of a neutral glycine molecule.
A denotes the atomic symbol C, N, or O.
The computational time per MD step was measured using
one CPU on a Sharp Mebius PCGP1C7U.
The energy cutoff of 113 (Ryd) was used for the numerical
integrations in all calculations. The results by the other theory
were taken from Ref. [15],
and the experimental values from Ref. [16].

SN 
TN 
TNDP 
SN 
SNP 
Other theory 
Expt. 

H4.0s1 
H4.0s3 
H4.0s3p2 
H4.0s31 
H4.0s31p21 
LDA/DZP 


A4.5s1p1 
A4.5s3p3 
A4.5s3p3d2 
A4.5s31p31 
A4.5s32p31d21 
Full potential 

(CC) 
1.555 
1.535 
1.528 
1.515 
1.528 
1.510 
1.532 
(NC) 
1.530 
1.480 
1.490 
1.502 
1.444 
1.439 
1.469 
(CO) 
1.353 
1.231 
1.235 
1.281 
1.238 
1.218 
1.207 
(CO) 
1.498 
1.365 
1.349 
1.416 
1.350 
1.348 
1.357 
(OH) 
1.144 
0.998 
0.987 
1.010 
0.995 
0.988 
 
(NCC) 
104.8 
106.5 
108.2 
108.4 
108.8 
114.8 
112.1 
(CCO) 
136.8 
127.9 
128.3 
128.9 
125.9 
124.9 
125.1 
(COH) 
96.8 
107.9 
105.7 
105.8 
106.8 
105.6 
 
C=ON 
3.132 
2.998 
2.905 
2.998 
2.882 
2.827 
 
Energy (Hartree) 
55.662 
55.981 
56.106 
55.818 
56.036 
 
 
Time(s)/MD step 
32 
86 
217 
34 
84 
 
 
Figure 1:
The radial wave function for of a carbon atom under the confinement
pseudo potential defined by Eq. (7), where 4.5 (a.u.), 4.3 (a.u.),
and 3.0 (Hartree) are used for ,
and , respectively.

Figure 2:
The total energy for a carbon dimer calculated using the eigenstates
as a basis set as a function of the number of basis orbitals per atom
for the cutoff radius of 3.5, 4.0, 4.5, 5.0, 5.5 and 6.0 (a.u.).

Figure 3:
The total energy for a carbon dimer C, a methane CH, and
the diamond as a function of the number of unoptimized (unopt) orbitals
and optimized orbitals with (rest) and without (unrest) the restriction.
The total energy and the number of orbitals are defined as those per
atom for C and the diamond, and as those per molecule for CH.
The energy cutoff of 113, 113, and 222 (Ryd) were used for the numerical
integrations in C, CH, and the diamond, respectively.
The two step convergence of C is due to the inclusion of dorbitals.

Figure 4:
The radial wave function of the minimal orbitals obtained by the
restricted optimization for the diamond and the lowest primitive
orbitals of a carbon atom. The optimization was done in the same
conditions as those in Fig. 3.

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