Lowest vertical excitation (S1) and detachment energies (D0, D n ) for model photoactive yellow protein (PYP) chromophores in the phenolate (left) and carboxylate (right) isomeric forms. The energies are in eV; these are the best estimates from Ref. 14 . The character of the resonance state is different in the two isomers. In the phenolate, where electron detachment from S1 to the lowest detachment continuum is a Koopmans-allowed one-electron transition, the excited state is a shape resonance. Carboxylate, in which the electron detachment from S1 to D0 is a Koopmans-forbidden two-electron process, is an example of a Feshbach resonance.
CIS calculations of the excited states of the phenolate form of the PYP chromophore. In a small basis set, which is not capable of representing continuum states, the ππ* transition (shown in red) appears as an isolated eigenstate and its energy approximates the position of the resonance. As the basis set is increased, numerous pseudo-continuum states appear below the resonance, making it more and more difficult to compute sufficiently large number of states such that the resonance is also included. 22 Moreover, the target state of interest begins to mix with pseudo-continuum states loosing its oscillator strength. In sum, standard excited-state methods are not capable of yielding converged (with respect to the basis set) positions of the auto-ionizing resonances and their lifetimes. The symmetry-decoupled Feshbach resonances, such as the ππ* state in the carboxylate form of PYP, are uncoupled from the continuum at the CIS level and their positions can be computed by standard approaches.
The transformation of the spectrum of the Hamiltonian upon complex scaling of all coordinates as described by the Balslev-Combes theorem. 31–33
θ-trajectories for the 2s2 Feshbach resonance in He shown on different scale. Angle θ varies from 0 to 0.500 rad (step 0.025 rad). 30s15p10d basis corresponds to even-tempered basis used in Ref. 43 , the gaussians' exponents values, α, vary within the range: 10−7 ⩽ α ⩽ 100, 2.66 × 10−4 ⩽ α ⩽ 30, and 2.66 × 10−4 ⩽ α ⩽ 30 for s, p, and d functions, respectively. 30s15p is the same basis without d-type basis functions. 20s10p5d (a) is formed from 30s15p10d by exclusion of the 10 s-, 5 p-, and 5 d-type most diffuse basis functions. 20s10p5d (b) even-tempered basis covers the same range of the gaussians' exponents, but with a greater scaling factor.
cs-EOM-EE-CCSD total energies for the ground and 1 S excited states of He (left panel) and H− (right panel). θ values corresponding to θ opt for the aug-cc-pVTZ+[10s5p5d] basis are 0.200 and 0.225 for He and H−, respectively. Three rays with the origin at the three lowest IEs of He and H− and rotated by the angle 2θ to the lower complex plain are shown in black. In the limit of the complete basis set the rays should coincide with the respective continuum branches.
θ-trajectories for the 2s2 Feshbach resonance in He shown on different scale. Angle θ varies from 0 to 0.500 rad (step 0.025 rad). See text for the diffuse subsets (3s, 3s3p, 3s3p3d, 10s5p5d) exponents definition.
θ-trajectories for the 2s2 Feshbach resonance in He. Angle θ varies from 0 to 0.500 rad (step 0.025 rad).
cs-EOM-EE-CCSD/aug-cc-pVTZ+[3s3p] electronic densities for the ground state (top) and 2s2 resonance (bottom) of He atom plotted in regular (left) and logarithmic (right) scale. Densities for θ = 0 and θ = 0.200 (real and imaginary parts, and the absolute value) are shown.
Decomposition of the He 2s2 resonance wave function into the excitations to diffuse orbitals (⟨R 2⟩ > 100 Å2, shown in blue), valence orbitals (⟨R 2⟩ < 100 Å2, shown in red), and mixed double excitations (green) for (a) aug-cc-pVTZ basis augmented with 3s, 3s3p, and 3s3p3d diffuse subsets; θ = 0.250; and for (b) θ = θ opt = 0.200 and θ = 0; aug-cc-pVTZ+[3s3p3d] basis set is used. Absolute values of amplitudes are used for the analysis.
Energy decomposition analysis for the ground state (top) and 2s2 resonance (bottom) of He. CS-EOM-EE-CCSD/aug-cc-pVTZ+[3s3p3d].
θ-trajectories for the 1s22p3s resonance in Be computed with cs-EOM-EE-CCSD/cs-CCSD/cs-HF and cs-EOM-EE-CCSD/cs-CCSD/HF using the 14s11p basis set from Ref. 81 .
Gaussian basis set exponents (α) of the first basis function and scaling factors (k, α i + 1 = α i /k) used in the even-tempered series.
Complex energies of the 2s2 resonance in helium calculated by cs-EOM-EE-CCSD in different bases.
Energies of the 2s2 resonance in H− calculated by cs-EOM-EE-CCSD in different bases. ΔE is given relative to the 1s ground state of neutral hydrogen computed for corresponding basis set.
Energies of the 1s22p3s resonance in Be calculated in different bases.
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