Theor Chem Acc (2015) 134:143
1 3
stable, faster convergence), but on the other hand it is more
complex and still Fermi-vacuum dependent.
Recently Sokolov and Chan published a paper where
another quasiparticle-based framework was outlined [ 40 ].
This approach has the enviable feature of Fermi-vacuum
independence. In that paper the concept of a non-particlenumber-conserving canonical transformation was introduced and, as a low-order approximation, the application of
a Bogoliubov transformation in a second-order perturbation
theory was investigated to describe MR situations. The presented results, as well as the intruder problem on the PES
in the BeH 2 model, indicate that the applied approximation
needs further improvements.
Although the original aim of our QMRCC project is
to develop tools to serve high accuracy calculations, it is
also worth investigating the possibility of a quasiparticlebased second-order perturbation theory as a low-cost alternative of the CC method. Development of such a perturbative approach would offer twofold benefi ts, namely it
would provide a cheap and easy to modify tool which
helps the development of the more complex CC approach
(understanding diffi culties, testing new ideas) and it can
be an effi cient alternative of the existing MR perturbation
theories. The aim of this paper is to investigate this second
point.
The early attempts to develop MR perturbation methods focused on the use of an effective Hamiltonian determined from the Bloch equation [ 7 , 18 , 24 , 27 ]. The main
drawback of these theories is the intruder state problem,
i.e., the appearance of close to zero denominators which
give nonphysical contribution to the effective Hamiltonian
matrix elements, especially for large CAS spaces where the
high lying model functions are energetically not separated
from the outer space determinants. To tackle with this problem the application of incomplete model spaces [ 18 , 27 ],
various level shift-based techniques [ 14 , 23 , 30 , 45 ] and
the concept of intermediate Hamiltonian [ 26 ] were intensively studied, but the most common solution is to use a
state-specifi c description, where only a single target state is
described [ 3 – 5 , 8 , 16 , 17 , 29 , 44 ].
Among the large set of various state-specifi c MR perturbation theories, we selected two popular methods to
compare our QMBPT2 approach to. These are the secondorder CAS-based PT (CASPT2) [ 2 , 3 , 8 ] method—which
is undoubtedly the most popular MR approach—and the
second-order n-electron-valence state PT (NEVPT2) [ 4 , 5 ]
method.
The zeroth-order Hamiltonian of the CASPT2 theory
is a projected generalized Fock operator, and the orthogonal functions are internally contracted excitations. In this
case the resolvent operator is not diagonal, and thus its
inverse is determined in an iterative process. To be able to
perform the inversion of the resolvent operator effi ciently,
the zero-order Hamiltonian has a block-diagonal structure. The well-known drawback of CASPT2 is the lack of
size-consistency.
The NEVPT approach has a list of remarkable qualitative properties (size-consistency, invariance to the rotation
of active orbitals, absence of intruder states, fi rst-order
correction to the wave function is a pure spin state) which
indicates that this as a serious candidate when an MR problem should be solve. Although with all these properties
the QMBPT2 method cannot compete (e.g., it is obviously
not invariant to the rotation of active orbitals), the relative
accuracy of these methods is still an interesting question.
The structure of the paper is as follows. In Sect. 2 the
quasiparticle approach is briefl y summarized, and the
new perturbation theory is introduced. The effi cient way
of implementation and the scaling properties will be presented in Sect. 3 . Some numerical results will be presented
in Sect. 4 and the conclusions of this study are collected in
Sect. 5 .
2 Theory
The main idea behind the new approach is to extend the
mathematical structure of the SR-based correlation methods to the MR case. To that end an orthonormal MR basis
is defi ned where one element of this basis is the reference
function which is supposed to a CAS wave function. We
also suppose that these functions can be uniquely labeled
by occupied and virtual one-particle indices as the ordinary
determinants, where these indices indicate the type of excitations with respect to a given predefi ned principal determinant (PD) hereafter denoted by |0 . In the notation of the
MR functions,
A, B, . . . and I, J, . . . letters are used for active virtual and
active occupied orbitals, respectively, while a, b, . . . and
i, j, . . . stand for the inactive occupied and inactive virtual
orbitals, fi nally Φ 0 is the reference CAS function.
Due to the one-particle labeling of these functions, we
have a mapping between these functions and any set of
determinants where a PD is defi ned and the excited determinants are labeled in a similar fashion. More precisely,
this mapping is a unitary transformation, where the PD is
transformed into the reference function Φ 0 = ˆ
U|0 , and
where | A
I , | a
i , . . . are ordinary determinants excited with
respect to the PD. Such a unitary transformation can be
(1)
Φ 0 , Φ
A
I , Φ
a
i , Φ
A
i , Φ
a
I , . . . , Φ
Ab
ij , Φ
ab
Ij , etc.
(2)
|Φ
A
I = ˆ
U|
A
I , |Φ
a
i = ˆ
U|
a
i , |Φ
A
i = ˆ
U|
A
i , . . . , |Φ
Ab
ij
= ˆ
U|
Ab
ij , |Φ
ab
Ij = ˆ
U|
ab
Ij , etc,
250
Reprinted from the journal
1 3
stable, faster convergence), but on the other hand it is more
complex and still Fermi-vacuum dependent.
Recently Sokolov and Chan published a paper where
another quasiparticle-based framework was outlined [ 40 ].
This approach has the enviable feature of Fermi-vacuum
independence. In that paper the concept of a non-particlenumber-conserving canonical transformation was introduced and, as a low-order approximation, the application of
a Bogoliubov transformation in a second-order perturbation
theory was investigated to describe MR situations. The presented results, as well as the intruder problem on the PES
in the BeH 2 model, indicate that the applied approximation
needs further improvements.
Although the original aim of our QMRCC project is
to develop tools to serve high accuracy calculations, it is
also worth investigating the possibility of a quasiparticlebased second-order perturbation theory as a low-cost alternative of the CC method. Development of such a perturbative approach would offer twofold benefi ts, namely it
would provide a cheap and easy to modify tool which
helps the development of the more complex CC approach
(understanding diffi culties, testing new ideas) and it can
be an effi cient alternative of the existing MR perturbation
theories. The aim of this paper is to investigate this second
point.
The early attempts to develop MR perturbation methods focused on the use of an effective Hamiltonian determined from the Bloch equation [ 7 , 18 , 24 , 27 ]. The main
drawback of these theories is the intruder state problem,
i.e., the appearance of close to zero denominators which
give nonphysical contribution to the effective Hamiltonian
matrix elements, especially for large CAS spaces where the
high lying model functions are energetically not separated
from the outer space determinants. To tackle with this problem the application of incomplete model spaces [ 18 , 27 ],
various level shift-based techniques [ 14 , 23 , 30 , 45 ] and
the concept of intermediate Hamiltonian [ 26 ] were intensively studied, but the most common solution is to use a
state-specifi c description, where only a single target state is
described [ 3 – 5 , 8 , 16 , 17 , 29 , 44 ].
Among the large set of various state-specifi c MR perturbation theories, we selected two popular methods to
compare our QMBPT2 approach to. These are the secondorder CAS-based PT (CASPT2) [ 2 , 3 , 8 ] method—which
is undoubtedly the most popular MR approach—and the
second-order n-electron-valence state PT (NEVPT2) [ 4 , 5 ]
method.
The zeroth-order Hamiltonian of the CASPT2 theory
is a projected generalized Fock operator, and the orthogonal functions are internally contracted excitations. In this
case the resolvent operator is not diagonal, and thus its
inverse is determined in an iterative process. To be able to
perform the inversion of the resolvent operator effi ciently,
the zero-order Hamiltonian has a block-diagonal structure. The well-known drawback of CASPT2 is the lack of
size-consistency.
The NEVPT approach has a list of remarkable qualitative properties (size-consistency, invariance to the rotation
of active orbitals, absence of intruder states, fi rst-order
correction to the wave function is a pure spin state) which
indicates that this as a serious candidate when an MR problem should be solve. Although with all these properties
the QMBPT2 method cannot compete (e.g., it is obviously
not invariant to the rotation of active orbitals), the relative
accuracy of these methods is still an interesting question.
The structure of the paper is as follows. In Sect. 2 the
quasiparticle approach is briefl y summarized, and the
new perturbation theory is introduced. The effi cient way
of implementation and the scaling properties will be presented in Sect. 3 . Some numerical results will be presented
in Sect. 4 and the conclusions of this study are collected in
Sect. 5 .
2 Theory
The main idea behind the new approach is to extend the
mathematical structure of the SR-based correlation methods to the MR case. To that end an orthonormal MR basis
is defi ned where one element of this basis is the reference
function which is supposed to a CAS wave function. We
also suppose that these functions can be uniquely labeled
by occupied and virtual one-particle indices as the ordinary
determinants, where these indices indicate the type of excitations with respect to a given predefi ned principal determinant (PD) hereafter denoted by |0 . In the notation of the
MR functions,
A, B, . . . and I, J, . . . letters are used for active virtual and
active occupied orbitals, respectively, while a, b, . . . and
i, j, . . . stand for the inactive occupied and inactive virtual
orbitals, fi nally Φ 0 is the reference CAS function.
Due to the one-particle labeling of these functions, we
have a mapping between these functions and any set of
determinants where a PD is defi ned and the excited determinants are labeled in a similar fashion. More precisely,
this mapping is a unitary transformation, where the PD is
transformed into the reference function Φ 0 = ˆ
U|0 , and
where | A
I , | a
i , . . . are ordinary determinants excited with
respect to the PD. Such a unitary transformation can be
(1)
Φ 0 , Φ
A
I , Φ
a
i , Φ
A
i , Φ
a
I , . . . , Φ
Ab
ij , Φ
ab
Ij , etc.
(2)
|Φ
A
I = ˆ
U|
A
I , |Φ
a
i = ˆ
U|
a
i , |Φ
A
i = ˆ
U|
A
i , . . . , |Φ
Ab
ij
= ˆ
U|
Ab
ij , |Φ
ab
Ij = ˆ
U|
ab
Ij , etc,
250
Reprinted from the journal