2 Advanced Electronic Structure Theory for High-Accuracy …
33
and discarding their invariance with respect to the XMS rotation. The state-specific
4-RDM is required as the highest-order RDM, and thus, the cu(4) approximation is
again exploited for DMRG-based (X)MS-CASPT2. In the XMS case, the approximate 4-RDM of the rotated reference ˜
L is calculated using the rotated low-order
RDMs ( ˜
L| ˆ
E
i
j | ˜
L, ˜
L| ˆ
E
ik
jl | ˜
L, and ˜
L| ˆ
E
ikm
jln | ˜
L) via the cumulant reconstruction formula instead of rotating the reconstructed 4-RDM.
Let us here biefly mention our computer implementation. The (X)MS-CASPT2
method with and without DMRG references was implemented into our in-house
program package. Derivations and implementations of the tensor contraction form
of the XMS-CASPT2 equations were achieved using the automated code generator
developed in Refs. [23, 24]. The quantum chemical DMRG program block [5] was
used to compute k-RDMs and k-TRDMs (k = 1, 2, 3) of multi-root (or multi-state)
DMRG wave functions [11]. In order to accelerate the evaluation of two-electron
integrals [28], the resolution-of-identity (or density-fitting) method was incorporated
into the (X)MS-CASPT2 implementation. The parallel implementation was deviced
by making effective use of the standard message-passing interface library and Global
Arrays (GA) toolkit [19] externally linked with the Aggregate Remote Memory Copy
Interface compatible library, ARMCI-MPI [6]. The memory allocations for twoelectron integrals and amplitudes are distributed across nodes to the data-parallel
arrays operated by the GA library. The computation of the amplitude equation (2.2)
is parallelized using GA’s data distribution functionality.
2.3 DMRG-XMS-CASPT2 Study of Diarylethene
Derivatives
In Ref. [29], we presented an application of DMRG-XMS-CASPT2 to the multistate calculations of the isomerization of the diarylethene derivatives, which are a
well-studied type of molecular photochromic systems [13]. Two kinds of the derivatives, (1) the 1,2-Bis(2-methyl-5-phenyl-3-thienyl)perfluorocyclopentene [12] and
(2) the 1,2-Bis(3-methyl-5-phenyl-2-thienyl)perfluorocyclopentene [27], denoted by
N-diarylethene and I-diarylethene, respectively, for short (Fig. 2.2) were examined.
They undergo a photo-switching cycloreversion transformation between two stable
isomers: open-ring and closed-ring.
Geometric parameters of the open- and closed-ring isomers were derived by the
geometry optimizations at the B3LYP-D3/def2-TZVPP level of theory for the singlet
ground state. The intermediate structures of the isomerization were derived by interpolating the two optimized open- and closed-ring geometries. The progress of the
isomerization is denoted as a function of the reaction coordinate F, ranging from 1
to 10, where the coordinates 1 and 10 correspond to the optimized open- and closedring isomers, respectively, denoted as iOF and iCF (i = 1 for N-diarylethene and
i = 2 for I-diarylethene), respectively.
33
and discarding their invariance with respect to the XMS rotation. The state-specific
4-RDM is required as the highest-order RDM, and thus, the cu(4) approximation is
again exploited for DMRG-based (X)MS-CASPT2. In the XMS case, the approximate 4-RDM of the rotated reference ˜
L is calculated using the rotated low-order
RDMs ( ˜
L| ˆ
E
i
j | ˜
L, ˜
L| ˆ
E
ik
jl | ˜
L, and ˜
L| ˆ
E
ikm
jln | ˜
L) via the cumulant reconstruction formula instead of rotating the reconstructed 4-RDM.
Let us here biefly mention our computer implementation. The (X)MS-CASPT2
method with and without DMRG references was implemented into our in-house
program package. Derivations and implementations of the tensor contraction form
of the XMS-CASPT2 equations were achieved using the automated code generator
developed in Refs. [23, 24]. The quantum chemical DMRG program block [5] was
used to compute k-RDMs and k-TRDMs (k = 1, 2, 3) of multi-root (or multi-state)
DMRG wave functions [11]. In order to accelerate the evaluation of two-electron
integrals [28], the resolution-of-identity (or density-fitting) method was incorporated
into the (X)MS-CASPT2 implementation. The parallel implementation was deviced
by making effective use of the standard message-passing interface library and Global
Arrays (GA) toolkit [19] externally linked with the Aggregate Remote Memory Copy
Interface compatible library, ARMCI-MPI [6]. The memory allocations for twoelectron integrals and amplitudes are distributed across nodes to the data-parallel
arrays operated by the GA library. The computation of the amplitude equation (2.2)
is parallelized using GA’s data distribution functionality.
2.3 DMRG-XMS-CASPT2 Study of Diarylethene
Derivatives
In Ref. [29], we presented an application of DMRG-XMS-CASPT2 to the multistate calculations of the isomerization of the diarylethene derivatives, which are a
well-studied type of molecular photochromic systems [13]. Two kinds of the derivatives, (1) the 1,2-Bis(2-methyl-5-phenyl-3-thienyl)perfluorocyclopentene [12] and
(2) the 1,2-Bis(3-methyl-5-phenyl-2-thienyl)perfluorocyclopentene [27], denoted by
N-diarylethene and I-diarylethene, respectively, for short (Fig. 2.2) were examined.
They undergo a photo-switching cycloreversion transformation between two stable
isomers: open-ring and closed-ring.
Geometric parameters of the open- and closed-ring isomers were derived by the
geometry optimizations at the B3LYP-D3/def2-TZVPP level of theory for the singlet
ground state. The intermediate structures of the isomerization were derived by interpolating the two optimized open- and closed-ring geometries. The progress of the
isomerization is denoted as a function of the reaction coordinate F, ranging from 1
to 10, where the coordinates 1 and 10 correspond to the optimized open- and closedring isomers, respectively, denoted as iOF and iCF (i = 1 for N-diarylethene and
i = 2 for I-diarylethene), respectively.