Geometry Optimization: Ver. 1.0
Taisuke Ozaki, RCIS, JAIST
The total energy of a system can be expanded by the Taylor series
with respect to atomic coordinates around with
as follows:
where the derivatives mean the partial derivatives at , and is the number of atoms.
By differentiating Eq. (1) with respect to , to the second order we have
In case the coordinates give a local minimum,
assuming
, we have the following
matrix equation:



(3) 
The short notation is



(4) 
where the matrix consisting of the second derivatives in the lefthand side
is called Hessian . Using Eq. (4), can be updated by



(5) 
This is the well known Newton method.
In the OpenMX, and in Eq. (5) are replaced by
and
given by the residual minimization
method in the direct inversion of iterative subspace (RMMDIIS) [1,2]
as follows:



(6) 
where is a tuning parameter for acceleration of the convergence,
which can be small (large) for a large (small)
.
in the RMMDIIS can be found by a linear combination of
previous upto pth gradients as



(7) 
where is found by minimizing
with
a constraint
.
According to Lagrange's multiplier method, is defined by
Considering
and
, an optimum set of can be
found by solving the following linear equation:



(9) 
An optimum choice of
may be obtained by the set of
coefficients as



(10) 
If the Hessian is approximated by the unity , Eq. (6) becomes



(11) 
This scheme in the Cartesian coordinate has been implemented
as 'DIIS' in OpenMX.
Define
Then, the BroydenFletcherGoldfarbShanno (BFGS)
method [3]
gives the following rank2 update formula for
:
where
.
An optimization scheme using Eq. (6) and the BFGS update formula
for the inverse of an approximate Hessian matrix in the Cartesian coordinate
has been implemented as 'BFGS' in OpenMX.
The BFGS update by Eq. (14) without any care
gives an illconditioned
approximate inverse of Hessian having negative eigenvalues
in many cases.
This leads to the optimization to saddle points rather than
the optimization to a minimum.
The rational function (RF) method [4]
can avoid the situation in principle. Instead of Eq. (1),
we may consider the following expression:
Then, the equation corresponding to Eq. (4) becomes



(16) 
Therefore, a large assures that
is positive definite.
If
is given by



(17) 
With Eq. (17), Eq. (16) may be equivalent to



(18) 
where the size of the matrix in the lefthand side is
, and called the augmented Hessian.
The lowest eigenvalue of the eigenvalue problem defined
by Eq. (18) may give an optimum choice for ,
and the corresponding eigenvector, the last component is
scaled to 1, gives an optimization step
.
In Eq. (18), the approximate Hessian can be estimated by
the following BFGS formula:
where .
An optimization scheme using Eq. (18) and the BFGS update formula
Eq. (19) in the Cartesian coordinate has been implemented
as 'RF' in OpenMX.
By diagonalizing the approximate Hessian given by Eq. (19),
the illconditioned situation can be largely reduced [5].
The approximate Hessian is diagonalized as



(20) 
where is a diagonal matrix of which diagonal parts
are eigenvalues of .
If the eigenvalue of the approximate Hessian is smaller than
a threshold (0.02 a.u. in OpenMX3.3), the eigenvalue is set to
the threshold.
The modification of eigenvalues gives a corrected matrix
instead of . Then, we have the inverse of
a corrected Hessian matrix being a positive definite as



(21) 
A optimization scheme using the inverse Eq. (21) in Eq. (6)
in the Cartesian coordinate has been implemented as 'EF' in OpenMX.
In addition, there are two important prescriptions for the stable
optimization:
(1)
If
is positive in the update of Hessian by Eq. (19), it is assured
that the updated Hessian is positive definite.
Therefore, if
is negative, the update should not be performed.
(2)
The maximum step should be always monitored, so that
an erratic movement of atomic position can be avoided.

 1

P. Csaszar and P. Pulay,
J. Mol. Struc. 114, 31 (1984).
 2

F. Eckert, P. Pulay, and H.J. Werner,
J. Comp. Chem. 18, 1473 (1997).
 3

C. G. Broyden, J. Inst. Math. Appl. 6, 76 (1970);
R. Fletcher, Comput. J. 13, 317 (1970);
D. Goldrarb, Math. Comp. 24, 23 (1970);
D. F. Shanno, Math. Comp. 24, 647 (1970).
 4

A. Banerjee et al., J. Phys. Chem. 89, 52 (1986).
 5

J. Baker, J. Comput. Chem. 7, 385 (1986).
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