1 3
Theor Chem Acc (2015) 134:143
DOI 10.1007/s00214-015-1746-z
REGULAR ARTICLE
A second-order multi-reference quasiparticle-based perturbation
theory
Zoltán Rolik
1 · Mihály Kállay
1
Received: 5 September 2015 / Accepted: 14 October 2015 / Published online: 7 November 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
The adequate description of electron correlation for molecular systems with multi-reference (MR) character is still a
challenge for the theory. Due to technical and theoretical
diffi culties—e.g., the polynomial scaling of computation
cost, treatment of large complete active spaces (CAS), the
question of extensivity—the applicability of existing multireference-based methods is limited to small systems. To
extend the scope of MR-based theories, a large number of
various formalisms emerged in this fi eld aiming at a proper
generalization for the successful single-reference-based
perturbation (PT) [ 6 , 28 ], coupled-cluster (CC) [ 9 – 11 ], and
confi guration interaction (CI) theories. An extensive review
of the MR-based CI and PT methods can be found in the
recent paper of Szalay et al. [ 42 ]. The authors of this paper
have also contributed to the improvement in these MR
methods [ 12 , 13 , 20 – 22 , 38 , 39 , 41 ] in many cases in cooperation with Péter Surján or with his support.
Along this line, in a recent paper [ 37 ] we introduced the
so-called quasiparticle-based MR CC method (QMRCC).
The mathematical structure of QMRCC is more or less the
same as that of the well-known SR CC theory, i.e., the reference function is a determinant, commuting cluster operators are applied, normal-ordering and diagram techniques
can be used, the method is extensive, etc. The point where
the MR description appears is the application of quasiparticle states instead of the ordinary molecular orbitals. These
quasiparticles are second-quantized many-particle objects
introduced by a unitary transformation which allows us to
represent the reference CAS function in a determinant-like
form. As it is shown in the cited paper, on one hand the
QMRCC method has some advantages with respect to the
closely related SR-based MR CC theory [ 22 , 31 , 34 ] (more
Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference
coupled-cluster approach (Rolik and Kállay in J Chem Phys
141:134112, 2014 ). The quasiparticles are introduced via a
unitary transformation which allows us to represent a complete active space reference function and other elements of
an orthonormal multi-reference basis in a determinant-like
form. The quasiparticle creation and annihilation operators
satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary
transformation these quasiparticles are also many-particle
objects, and the Hamilton operator in the quasiparticle
basis contains higher than two-body terms. The defi nition
of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several
of its benefi cial properties like the extensivity. The effi cient
implementation of the method is briefl y discussed, and test
results are also presented.
Keywords Multi-reference · Perturbation theory ·
Quasiparticles
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Zoltán Rolik
rolik@mail.bme.hu
1
MTA-BME “Lendület” Quantum Chemistry Research
Group, Department of Physical Chemistry and Materials
Science , Budapest University of Technology and Economics ,
H1521 Budapest , Hungary
249
Reprinted from the journal
Theor Chem Acc (2015) 134:143
DOI 10.1007/s00214-015-1746-z
REGULAR ARTICLE
A second-order multi-reference quasiparticle-based perturbation
theory
Zoltán Rolik
1 · Mihály Kállay
1
Received: 5 September 2015 / Accepted: 14 October 2015 / Published online: 7 November 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
The adequate description of electron correlation for molecular systems with multi-reference (MR) character is still a
challenge for the theory. Due to technical and theoretical
diffi culties—e.g., the polynomial scaling of computation
cost, treatment of large complete active spaces (CAS), the
question of extensivity—the applicability of existing multireference-based methods is limited to small systems. To
extend the scope of MR-based theories, a large number of
various formalisms emerged in this fi eld aiming at a proper
generalization for the successful single-reference-based
perturbation (PT) [ 6 , 28 ], coupled-cluster (CC) [ 9 – 11 ], and
confi guration interaction (CI) theories. An extensive review
of the MR-based CI and PT methods can be found in the
recent paper of Szalay et al. [ 42 ]. The authors of this paper
have also contributed to the improvement in these MR
methods [ 12 , 13 , 20 – 22 , 38 , 39 , 41 ] in many cases in cooperation with Péter Surján or with his support.
Along this line, in a recent paper [ 37 ] we introduced the
so-called quasiparticle-based MR CC method (QMRCC).
The mathematical structure of QMRCC is more or less the
same as that of the well-known SR CC theory, i.e., the reference function is a determinant, commuting cluster operators are applied, normal-ordering and diagram techniques
can be used, the method is extensive, etc. The point where
the MR description appears is the application of quasiparticle states instead of the ordinary molecular orbitals. These
quasiparticles are second-quantized many-particle objects
introduced by a unitary transformation which allows us to
represent the reference CAS function in a determinant-like
form. As it is shown in the cited paper, on one hand the
QMRCC method has some advantages with respect to the
closely related SR-based MR CC theory [ 22 , 31 , 34 ] (more
Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference
coupled-cluster approach (Rolik and Kállay in J Chem Phys
141:134112, 2014 ). The quasiparticles are introduced via a
unitary transformation which allows us to represent a complete active space reference function and other elements of
an orthonormal multi-reference basis in a determinant-like
form. The quasiparticle creation and annihilation operators
satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary
transformation these quasiparticles are also many-particle
objects, and the Hamilton operator in the quasiparticle
basis contains higher than two-body terms. The defi nition
of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several
of its benefi cial properties like the extensivity. The effi cient
implementation of the method is briefl y discussed, and test
results are also presented.
Keywords Multi-reference · Perturbation theory ·
Quasiparticles
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Zoltán Rolik
rolik@mail.bme.hu
1
MTA-BME “Lendület” Quantum Chemistry Research
Group, Department of Physical Chemistry and Materials
Science , Budapest University of Technology and Economics ,
H1521 Budapest , Hungary
249
Reprinted from the journal