Multistate Complete-Active-Space Second-Order Perturbation Theory Based on Density Matrix Renormalization Group Reference States

Abstract

We present the development of the multistate multireference second-order perturbation theory (CASPT2) with multiroot references, which are described using the density matrix renormalization group (DMRG) method to handle a large active space. The multistate first-order wave functions are expanded into the internally contracted (IC) basis of the single-state single-reference (SS-SR) scheme, which is shown to be the most feasible variant to use DMRG references. The feasibility of the SS-SR scheme comes from two factors: first, it formally does not require the fourth-order transition reduced density matrix (TRDM) and second, the computational complexity scales linearly with the number of the reference states. The extended multistate (XMS) treatment is further incorporated, giving suited treatment of the zeroth-order Hamiltonian despite the fact that the SS-SR based IC basis is not invariant with respect to the XMS rotation. In addition, the state-specific fourth-order reduced density matrix (RDM) is eliminated in an approximate fashion using the cumulant reconstruction formula, as also done in the previous state-specific DMRG-cu(4)-CASPT2 approach. The resultant method, referred to as DMRG-cu(4)-XMS-CASPT2, uses the RDMs and TRDMs of up to third-order provided by the DMRG calculation. The multistate potential energy curves of the photoisomerization of diarylethene derivatives with CAS(26e,24o) are presented to illustrate the applicability of our theoretical approach.

A comprehensive benchmark of the XMS-CASPT2 method for the photochemistry of a retinal chromophore model

“…This shift of the CI location towards the TSDIR ( Such artefacts are significantly reduced by using the SS-SR contraction scheme instead of MSMR ( Figure S1). Similar observation in favour of the SS-SR contraction was reported for the MS-CASPT2 method and attributed primarily due to the highly mixed nature of the CASSCF reference wave functions [26]. The extended multireference method is considered to be a remedy for this problem and indeed as shown in Figure 2, the XMS-CASPT2 energy profiles are free of artefacts.…”

“…In order to avoid this steep increase in the number of excited determinants, an internal contraction scheme was implemented and successfully applied to the CASPT2 method by Roos and coworkers [1,3,4]. In multi-state (MS) CASPT2 formalism, two types of the internal contraction schemes exist, namely single-state single reference (SS-SR) and multi-state multireference (MS-MR) [25][26][27]. In SS-SR, each state is computed independently before mixing and the contracted reference state is specific for the state under consideration.…”

Magnetic circular dichroism of an unaromatic planar [8]annulene

“…Using this geometry for D 8 h 1 and DFT‐optimized geometries for D 4 h 1 and 2 , vertical excitation energies and absorption and MCD intensities of 1 and 2 were calculated by 2 methods: (1) multistate complete active space second‐order perturbation theory (MS‐CASPT2) and (2) an extended version of MS‐CASPT2 (extended multistate complete active space second‐order perturbation theory [XMS‐CASPT2]), in which an extra rotation of reference CASSCF states, capable of accounting for off‐diagonal zeroth‐order coupling, is performed prior to solution of the CASPT2 equations (XMS‐CASPT2 uses different reference states than MS‐CASPT2). Details of the XMS‐CASPT2 implementation used have been published elsewhere . MCD calculations were performed with the previously proposed first‐principles method, in which the Zeeman terms between the singlet electronic states are explicitly included via quasi‐degenerate perturbation theory in the form β e L · B , where β e is Bohr magneton, L is angular momentum, and B is magnetic field.…”

“…Details of the XMS-CASPT2 implementation used have been published elsewhere. [44] MCD calculations were performed with the previously proposed [45] first-principles method, in which the Zeeman terms between the singlet electronic states are explicitly included via quasi-degenerate perturbation theory in the form β e L·B, where β e is Bohr magneton, L is angular momentum, and B is magnetic field. Note that all components of the degenerate states are involved and that the approach yields Zeemanperturbed wave functions, for which the total "all-term" MCD intensity is calculated.…”

Multireference Approaches for Excited States of Molecules

“…cu(4) approximation, as has been done in conjunction with statespecific CASPT2, 459,460 NEVPT2, 455 and MRCI. 70 Extension in a different direction is exampled by developing of a multistate version of the DMRG-CASPT2 method by Yanai et al 461 DMRG-based methods have been used to calculate the lowest excited states of small systems LiF, 461,462 CH2, 463 HNCO, 464 C2, 465 N2, 466 Cr2, 466 ethylene, 457,467 H2O, 451,466 CsH, 468 CoH, 469 Fe2S2 , 435 NiCO, 470 a prototype of carbonyl metal complex, and moderate-sized molecules, e.g. uracil 471 and indole 466 to demonstrate efficiency of implemented algorithms.…”