Evaluation of a two RDM element . We can obtain this element, e.g., at the block configuration where indices 4 and 1 are on the left block and indices 6, 7 are on the right block [corresponding to calling COMPUTE(2, 0, 2) in Algorithm 1].
DMRG-CASSCF excitation energies in eV for the , , and states in the conjugated polyenes to .
Converged DMRG sweep energies in Hartrees vs number of orbital optimization macroiterations in .
Change in CASSCF energies of the low-lying states of as a function of increasing the active space from (4,4) to (12,12) (i.e., complete valence active space).
DMRG-CASSCF excitation energies for the low-lying singlet excited states of polyenes ranging from to . The excitation energies are plotted against where is the number of double bonds. The ratio of the slopes for the different states is found to be 2:3.0:3.8 as compared to 2:3.1:3.8 experimentally. Inset: The same plot for the CASCI-MRMP excitation energies from Kurashige et al. (Ref. 43). As can be seen, these show a different and less linear dependence on .
Polyene and carotene excitation energies vs the number of double bonds: The -carotene excitation energies when fitted to the polyene excitation energies give an effective conjugation length of 9.5–9.7.
Natural orbitals corresponding to the through states. These orbitals participate in the lowest-lying singlet excitations in -carotene and contain little density on the nonplanar end groups.
ALGORITHM 1. Two-particle density matrix evaluation showing how the two-particle density matrix is assembled across a DMRG sweep.
ALGORITHM 2. COMPUTE(nl, np, nr, left, sitep, right) Note and , i.e., the number of indices in the two-particle density matrix .
ALGORITHM 3. Parallel four-index integral transformation algorithm.
Energies, symmetries, and oscillator strengths for the lowest-lying singlet excited states in conjugated polyenes. The DMRG-PAO-CASCI and DMRG-CASSCF entries for the ground states give the total energy in ; the other entries give the excitation energy from the ground state in eV. The estimated error of the DMRG-CASSCF energies from the exact CASSCF energies in the same active space is less than . The notation denotes the active space used in the DMRG-PAO-CASCI and DMRG-CASSCF calculations. Oscillator strengths are in a.u. for the ground state, excited state transitions. The CASCI-MRMP excitation energies are from Kurashige et al. (Ref. 43); note that these used at most a (10,10) active space. The experimental numbers in brackets are from measurements on the substituted polyene, spheroidene (Ref. 72).
Single particle nature of the polyene excitations (in %). For a given excited state (e.g., ), the excitation weight of the transition is given by . The total excitation weight is the sum of weights for all transition; 100% indicates that the given excited state corresponds entirely to single excitations from the ground state. The transition labels are interpreted as follows: 1, 2, 3,…denote HOMO, , , etc., natural orbitals, while , , and denote LUMO, , and natural orbitals. As the polyenes increase in length, the total weight of the single excitations in the low-lying states becomes very small, .
DMRG-CASSCF energies, symmetries, and oscillator strengths for the lowest-lying singlet excited states in -carotene with the complete -valence (22,22) active space. Total energies in , excitation energies in eV, and oscillator strengths in a.u. The estimated error of the DMRG-CASSCF energies from the exact CASSCF energies in the same active space is less than . Oscillator strengths are for the ground state, excited state transitions.
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