1 3
Theor Chem Acc (2015) 134:100
DOI 10.1007/s00214-015-1703-x
REGULAR ARTICLE
Novel orthogonalization and biorthogonalization algorithms
Towards multistate multiconfi guration perturbation theory
Zsuzsanna Tóth
1 · Péter R. Nagy
2 · Péter Jeszenszki
1 · Ágnes Szabados
1
Received: 27 May 2015 / Accepted: 15 July 2015 / Published online: 1 August 2015
© Springer-Verlag Berlin Heidelberg 2015
nonorthogonal set of N -dimensional vectors. Arranging vectors
c i as columns in matrix C , the overlap matrix
is a positive defi nite, nonunit matrix. Orthonormalization
implies a linear transformation B = CL o , obeying
Vector set
c i N
i=1
, biorthonormal to
c i N
i=1
is obtained by the
transformation
C = CL bo , fulfi lling
where
C contains vectors c i in columns. In contrast to orthonormalization, the task of biorthonormal or reciprocal set construction has the uniquely defi ned solution
as a direct consequence of Eq. ( 2 ).
Of the possible orthonormalization procedures, Löwdin’s symmetric scheme [ 1 , 2 ], operating with L o = S −1/2 ,
is widely exploited in quantum chemistry. Gram–Schmidt
orthogonalization is a popular, less costly alternative
that lacks the symmetry conservation and resemblance
[ 2 – 4 ] properties of Löwdin’s symmetric treatment. Gram–
Schmidt orthogonalization is known to depend on the
ordering of vectors c i , which is not necessarily a shortcoming. It is a deliberate advantage, e.g., if one, selected vector
is meant to be fi xed.
This situation may be met when aiming to describe
dynamic electron correlation starting from a single, multideterminantal reference vector. Correction schemes based
on perturbation theory (PT) have been applying successive
Gram–Schmidt orthogonalization in such circumstances [ 5 –
7 ], occasionally combined with Löwdin’s symmetrical [ 8 ] or
S = C
† C
(1)
B
† B = L
†
o SL o = I .
(2)
C
† C = L
†
bo S = I ,
L bo = S
−1
Abstract Orthogonalization with the prerequisite of
keeping several vectors fi xed is examined. Explicit formulae are derived both for orthogonal and biorthogonal vector
sets. Calculation of the inverse or square root of the entire
overlap matrix is eliminated, allowing computational time
reduction. In this special situation, it is found suffi cient to
evaluate the functions of matrices of the dimension matching the number of fi xed vectors. The (bi)orthogonal sets
fi nd direct application in extending multiconfi gurational
perturbation theory to deal with multiple reference vectors.
Keywords Overlap · Orthogonalization · Biorthogonal
sets · Multiconfi guration perturbation theory · Multistate
theory
1 Introduction
There are two different, equivalent approaches for treating nonorthogonality of a nonredundant vector set in a linear algebraic
problem. One way is creating biorthogonal vectors to the overlapping set, and the other more common way is orthogonalizing the basis set. Let us assume that
c i N
i=1
is a nonredundant,
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Ágnes Szabados
szabados@chem.elte.hu
1
Laboratory of Theoretical Chemistry , Eötvös University ,
POB 32 , Budapest 1518 , Hungary
2
MTA-BME Lendület Quantum Chemistry Research Group,
Department of Physical Chemistry and Materials Science ,
Budapest University of Technology and Economics , POB 91 ,
Budapest 1521 , Hungary
229
Reprinted from the journal
Theor Chem Acc (2015) 134:100
DOI 10.1007/s00214-015-1703-x
REGULAR ARTICLE
Novel orthogonalization and biorthogonalization algorithms
Towards multistate multiconfi guration perturbation theory
Zsuzsanna Tóth
1 · Péter R. Nagy
2 · Péter Jeszenszki
1 · Ágnes Szabados
1
Received: 27 May 2015 / Accepted: 15 July 2015 / Published online: 1 August 2015
© Springer-Verlag Berlin Heidelberg 2015
nonorthogonal set of N -dimensional vectors. Arranging vectors
c i as columns in matrix C , the overlap matrix
is a positive defi nite, nonunit matrix. Orthonormalization
implies a linear transformation B = CL o , obeying
Vector set
c i N
i=1
, biorthonormal to
c i N
i=1
is obtained by the
transformation
C = CL bo , fulfi lling
where
C contains vectors c i in columns. In contrast to orthonormalization, the task of biorthonormal or reciprocal set construction has the uniquely defi ned solution
as a direct consequence of Eq. ( 2 ).
Of the possible orthonormalization procedures, Löwdin’s symmetric scheme [ 1 , 2 ], operating with L o = S −1/2 ,
is widely exploited in quantum chemistry. Gram–Schmidt
orthogonalization is a popular, less costly alternative
that lacks the symmetry conservation and resemblance
[ 2 – 4 ] properties of Löwdin’s symmetric treatment. Gram–
Schmidt orthogonalization is known to depend on the
ordering of vectors c i , which is not necessarily a shortcoming. It is a deliberate advantage, e.g., if one, selected vector
is meant to be fi xed.
This situation may be met when aiming to describe
dynamic electron correlation starting from a single, multideterminantal reference vector. Correction schemes based
on perturbation theory (PT) have been applying successive
Gram–Schmidt orthogonalization in such circumstances [ 5 –
7 ], occasionally combined with Löwdin’s symmetrical [ 8 ] or
S = C
† C
(1)
B
† B = L
†
o SL o = I .
(2)
C
† C = L
†
bo S = I ,
L bo = S
−1
Abstract Orthogonalization with the prerequisite of
keeping several vectors fi xed is examined. Explicit formulae are derived both for orthogonal and biorthogonal vector
sets. Calculation of the inverse or square root of the entire
overlap matrix is eliminated, allowing computational time
reduction. In this special situation, it is found suffi cient to
evaluate the functions of matrices of the dimension matching the number of fi xed vectors. The (bi)orthogonal sets
fi nd direct application in extending multiconfi gurational
perturbation theory to deal with multiple reference vectors.
Keywords Overlap · Orthogonalization · Biorthogonal
sets · Multiconfi guration perturbation theory · Multistate
theory
1 Introduction
There are two different, equivalent approaches for treating nonorthogonality of a nonredundant vector set in a linear algebraic
problem. One way is creating biorthogonal vectors to the overlapping set, and the other more common way is orthogonalizing the basis set. Let us assume that
c i N
i=1
is a nonredundant,
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan.”
* Ágnes Szabados
szabados@chem.elte.hu
1
Laboratory of Theoretical Chemistry , Eötvös University ,
POB 32 , Budapest 1518 , Hungary
2
MTA-BME Lendület Quantum Chemistry Research Group,
Department of Physical Chemistry and Materials Science ,
Budapest University of Technology and Economics , POB 91 ,
Budapest 1521 , Hungary
229
Reprinted from the journal