134
A. Łachma´ nska et al.
which scales binomially with system size, has been replaced by a product of lowerdimensional tensors.
The DMRG algorithm is a powerful tool to approximate FCI wave functions
in a given active space that are computationally not accessible for conventional
quantum mechanical methods like CASSCF. Most importantly, it is suitable for
strongly-correlated systems and hence allows us to accurately model heavy-element
compounds, like transition metal- or actinide-containing molecules. Although originally formulated to tackle one-dimensional problems, DMRG has been successfully
applied to describe strong correlation in general 3-dimensional systems. The missing
dynamical correlation effects can be added a posteriori using the same corrections
as developed for traditional MCSCF methods. Examples are second order complete active space perturbation theory (DMRG-CASPT2), [82] the multi-reference
configuration interaction approach with internal contraction of DMRG (DMRGicMRCI), [128] canonical transformation theory (CT), [106] or perturbation theory
formulated in terms of matrix product states [132].
3.2.6 Geminal-Based Approaches
All electronic structure methods discussed above use one-electron functions (orbitals)
to construct the Slater determinants that span the N -particle Hilbert space. A conceptually different approach to account for electron correlation effects is to use twoelectron functions as fundamental building blocks of the electronic wave function [21,
73, 96]. In second quantization, a (singlet) two-electron operator ψ
†
i , also called geminal, can be written as a linear combinations of electron-pair creators,
ψ
†
i =
M
q=1
C
i
q a
†
q a
†
¯
q ,
(23)
where a
†
q (a
†
¯
q ) are the fermionic creation operators for α (β) electrons and M is the
number of one-electron functions used to construct geminal i. In the above equation,
C
i
q are the geminal coefficients that link the geminal creation operator with the
underlying one-particle basis (represented by a
†
p ). Thus, geminals are quasi-particles
and the corresponding geminal creation operators are electron pair creators. The
geminal-based electronic wave function is a product of the geminal creation operators
acting on the vacuum state,
Ψ el =
P
i
ψ
†
i
,
(24)
with P = N /2 being the number of electron pairs. If the geminal creation operators
have the general form of (23), we obtain the antisymmetric product of interacting
geminals (APIG) wave function [96]. Although APIG includes correlations between
orbital pairs, it is computationally intractable for larger systems. Substituting (23) in
A. Łachma´ nska et al.
which scales binomially with system size, has been replaced by a product of lowerdimensional tensors.
The DMRG algorithm is a powerful tool to approximate FCI wave functions
in a given active space that are computationally not accessible for conventional
quantum mechanical methods like CASSCF. Most importantly, it is suitable for
strongly-correlated systems and hence allows us to accurately model heavy-element
compounds, like transition metal- or actinide-containing molecules. Although originally formulated to tackle one-dimensional problems, DMRG has been successfully
applied to describe strong correlation in general 3-dimensional systems. The missing
dynamical correlation effects can be added a posteriori using the same corrections
as developed for traditional MCSCF methods. Examples are second order complete active space perturbation theory (DMRG-CASPT2), [82] the multi-reference
configuration interaction approach with internal contraction of DMRG (DMRGicMRCI), [128] canonical transformation theory (CT), [106] or perturbation theory
formulated in terms of matrix product states [132].
3.2.6 Geminal-Based Approaches
All electronic structure methods discussed above use one-electron functions (orbitals)
to construct the Slater determinants that span the N -particle Hilbert space. A conceptually different approach to account for electron correlation effects is to use twoelectron functions as fundamental building blocks of the electronic wave function [21,
73, 96]. In second quantization, a (singlet) two-electron operator ψ
†
i , also called geminal, can be written as a linear combinations of electron-pair creators,
ψ
†
i =
M
q=1
C
i
q a
†
q a
†
¯
q ,
(23)
where a
†
q (a
†
¯
q ) are the fermionic creation operators for α (β) electrons and M is the
number of one-electron functions used to construct geminal i. In the above equation,
C
i
q are the geminal coefficients that link the geminal creation operator with the
underlying one-particle basis (represented by a
†
p ). Thus, geminals are quasi-particles
and the corresponding geminal creation operators are electron pair creators. The
geminal-based electronic wave function is a product of the geminal creation operators
acting on the vacuum state,
Ψ el =
P
i
ψ
†
i
,
(24)
with P = N /2 being the number of electron pairs. If the geminal creation operators
have the general form of (23), we obtain the antisymmetric product of interacting
geminals (APIG) wave function [96]. Although APIG includes correlations between
orbital pairs, it is computationally intractable for larger systems. Substituting (23) in