30
T. Yanai
derived by incorporating the density matrix renormalization group (DMRG) wave
functions into the multi-state reference for the XMS-CASPT2 calculations.
The CASPT2 theory is a most widely used MR method because of its low computational cost and theoretical simplicity [1, 2, 7]. In the CASPT2 calculations,
the excited states of interest are first accounted for in a qualitative fashion by the
complete active-space self-consistent field (CASSCF) wave functions [21, 22]. As
a well-established manner, the state-averaged (SA) CASSCF treatment is used to
obtain the references as a multi-root solution that provides a ground state and several low-lying excited states. Using these CASSCF states as the zeroth-order, the
CASPT2 method describes the first-order perturbed wave functions and secondorder perturbation energies to further account for quantitative correlation to obtain
chemical accuracy, as well as in practice even qualitative correctness, including state
ordering.
In order to use CASPT2 for excited states, its state-specific variant is well known
to be inadequate for studying near-degenerate states, in which the strong mixing
arises between the reference and other secondary CASSCF states via the secondorder interactions. Another issue is a technical difficulty in determining state-specific
CASSCF references of excited states because of instability of convergence causing
so-called root flopping oscillation. These issues can be properly addressed by the
multi-state treatment based on state-average CASSCF references, known as the MSCASPT2 theory [7], an extension of CASPT2 formulated on basis of the quasidegenerate perturbation theory (QDPT) [4]. It constructs an effective Hamiltonian
matrix in the reference state basis under the presence of the perturbation, and its
diagonalization yields the mixing of the perturbed states. This procedure allows us
to properly describe the state crossing. The MS extension of MR perturbation theory
was similarly developed in the MCQDPT [17, 18] and QD-NEVPT2 [3] methods.
The combination of Granovsky’s extended MS (XMS) method and the MS-CASPT2
method [10] was developed by Shiozaki et al. to improve the zeroth-order description.
This approach (XMS-CASPT2) is able to additionally account for the off-diagonal
Fock operator-based coupling [26].
2.2 Overview of DMRG-XMS-CASPT2 Theory
In this section, the multi-state (MS) extension of the CASPT2 is briefly reviewed. In
the main framework of CASPT2, the first-order wave function is expanded into the
internally contracted (IC) basis, which is generated by applying excitation operators
to the zeroth-order reference. The MS treatment allows us to have multiple zerothorder references resulting from the preceding state-average CASSCF calculations.
The IC basis construction shown here is based on the single-state single-reference
(SS-SR) contraction. As done in the state-specific CASPT2 theory, the IC configurations for the first-order space are generated solely relative to each of CAS-CI/SCF
states {|L}. The SS-SR scheme is expressed with the following ansatz:
T. Yanai
derived by incorporating the density matrix renormalization group (DMRG) wave
functions into the multi-state reference for the XMS-CASPT2 calculations.
The CASPT2 theory is a most widely used MR method because of its low computational cost and theoretical simplicity [1, 2, 7]. In the CASPT2 calculations,
the excited states of interest are first accounted for in a qualitative fashion by the
complete active-space self-consistent field (CASSCF) wave functions [21, 22]. As
a well-established manner, the state-averaged (SA) CASSCF treatment is used to
obtain the references as a multi-root solution that provides a ground state and several low-lying excited states. Using these CASSCF states as the zeroth-order, the
CASPT2 method describes the first-order perturbed wave functions and secondorder perturbation energies to further account for quantitative correlation to obtain
chemical accuracy, as well as in practice even qualitative correctness, including state
ordering.
In order to use CASPT2 for excited states, its state-specific variant is well known
to be inadequate for studying near-degenerate states, in which the strong mixing
arises between the reference and other secondary CASSCF states via the secondorder interactions. Another issue is a technical difficulty in determining state-specific
CASSCF references of excited states because of instability of convergence causing
so-called root flopping oscillation. These issues can be properly addressed by the
multi-state treatment based on state-average CASSCF references, known as the MSCASPT2 theory [7], an extension of CASPT2 formulated on basis of the quasidegenerate perturbation theory (QDPT) [4]. It constructs an effective Hamiltonian
matrix in the reference state basis under the presence of the perturbation, and its
diagonalization yields the mixing of the perturbed states. This procedure allows us
to properly describe the state crossing. The MS extension of MR perturbation theory
was similarly developed in the MCQDPT [17, 18] and QD-NEVPT2 [3] methods.
The combination of Granovsky’s extended MS (XMS) method and the MS-CASPT2
method [10] was developed by Shiozaki et al. to improve the zeroth-order description.
This approach (XMS-CASPT2) is able to additionally account for the off-diagonal
Fock operator-based coupling [26].
2.2 Overview of DMRG-XMS-CASPT2 Theory
In this section, the multi-state (MS) extension of the CASPT2 is briefly reviewed. In
the main framework of CASPT2, the first-order wave function is expanded into the
internally contracted (IC) basis, which is generated by applying excitation operators
to the zeroth-order reference. The MS treatment allows us to have multiple zerothorder references resulting from the preceding state-average CASSCF calculations.
The IC basis construction shown here is based on the single-state single-reference
(SS-SR) contraction. As done in the state-specific CASPT2 theory, the IC configurations for the first-order space are generated solely relative to each of CAS-CI/SCF
states {|L}. The SS-SR scheme is expressed with the following ansatz: