Excited-State Spectra of Strongly Correlated Molecules from a

Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 5373−5378

pubs.acs.org/JPCL

Excited-State Spectra of Strongly Correlated Molecules from a Reduced-Density-Matrix Approach S. Hemmatiyan, M. Sajjan, A. W. Schlimgen, and D. A. Mazziotti* Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, United States

J. Phys. Chem. Lett. Downloaded from pubs.acs.org by ST FRANCIS XAVIER UNIV on 09/05/18. For personal use only.

S Supporting Information *

ABSTRACT: Excited-state energies are computed in the space of single-electron transitions from the ground state from only a knowledge of the two-electron reduced density matrix (2-RDM). Previous work developed and applied the theory to small molecular systems with accurate results, but applications to both larger and more correlated molecules were hindered by ill-conditioning of the effective eigenvalue problem. Here we improve the excited-spectra 2-RDM theory through a stable Hamiltonian-shifted regularization algorithm that removes the near singularities within the computation. The theory with ground-state 2-RDMs from the variational 2-RDM method is applied to the excited energies of strongly correlated molecules including the optical band gap of hydrogen and acene chains, the singlet−triplet splitting of nickel dithiolates, as well as the low-lying excited states of an optical dye. While single-excitation theories like CISD and TD-DFT underestimate band gaps and excited-state splittings, the 2-RDM theory yields band gap and excited-state splittings that are in good agreement with full configuration interaction and experiment where available.

E

from ground-state 2-RDMs that are obtainable directly from 2RDM methods as well as from wave function methods. The ES-2RDM theory is applied to problems involving strongly correlated molecules including determination of the optical band gap of hydrogen and acene chains, the computation of singlet−triplet splitting of nickel dithiolates, as well as the calculation of low-lying excited-state energies of an optical dye (see Figure 1). The strong correlation is captured through calculation of the ground-state 2-RDM from the variational 2-RDM (v2-RDM) method in which the energy is minimized with the 2-RDM constrained by N-representability conditions, which are necessary for the 2-RDM to represent an N-electron quantum state. Single-excitation theories like CISD and TD-DFT are known to underestimate band gaps and excited-state splittings, resulting in the production of red-shifted spectra for an overly metallic description of molecules and materials. While the lowest approximation in the ES-2RDM theory is also based on single excitations, its single excitations are from the correlated 2-RDM ground state, harnessing all of the strong electron correlation captured by the v2-RDM method. In the applications studied here, the ES-2RDM method yields band gap and excited-state splittings that are in good agreement with those from full configuration interaction and experiment where available. The mth excited-state energy can be approximated from

xcited-state energies and properties are critically important for the prediction of chemical transformations, especially in the case of photoexcited processes.1−3 Extensions of ground-state methods have been developed to compute excited-state spectra from ground-state information. Examples include configuration interaction singles (CISD),4 timedependent density functional theory (TD-DFT),5−7 and the equation-of-motion coupled cluster (EOM-CC) methods.8 These methods consider single- or single-and-double-electron excitations from the mean-field wave function or one-electron density. Although such methods work well for ground and excited states near the mean-field limit, they can have difficulty describing strongly correlated states with consequences for the description of important phenomena like conical intersections and metal-to-insulator transitions. In this Letter, we improve a theory for the computation of excited-state spectra from any ground-state two-electron reduced density matrix (2-RDM).9−18 In the excited-spectra RDM theory known as the Hermitian operator method,9−13,15,16 the excited-state energies are computed in the space of p-electron transitions from the correlated ground-state wave function from a knowledge of only the 2p-RDM. Previous work developed and applied the theory for p = 1 to small molecular systems with accurate results,12,13,15,17 but applications to both larger and more correlated molecules were hindered by ill-conditioning of the effective eigenvalue problem. Here we improve a family of excited-spectra 2RDM (ES-2RDM) theories through a stable Hamiltonianshifted regularization algorithm that removes the near singularities within the computation. The resulting 2-RDM theory is applicable to computing the excited-state spectra © XXXX American Chemical Society

Received: August 9, 2018 Accepted: August 31, 2018

5373

DOI: 10.1021/acs.jpclett.8b02455 J. Phys. Chem. Lett. 2018, 9, 5373−5378

Letter

The Journal of Physical Chemistry Letters

Although these two reductions introduce approximations for approximate ground-state 2p-RDMs, previous work12 has shown that for single particle−hole transitions (p = 1) this combination of approximations, known as the Hermitian operator method, depends only upon the 2-RDM and produces energy spectra that capture single excitations, renormalized relative to their correlated ground states. The explicit 2-RDM dependence of the effective Hamiltonian in the numerator and the particle−hole RDM in the denominator of eq 4 is presented in ref 12. Because the ground-state 2-RDMs can be correlated, the p = 1 approximation, in contrast to single excitation approximations like CIS or TDDFT, can generate excited states that are accurate single-electron excitations of strongly correlated electronic ground states. Many low-lying excited sates are well-represented by only considering single-particle excitations extracted from the exact ground state.12,14 As discussed in refs 12 and 15, for excited-spectra RDM methods that depend on higher-order RDMs, either through the use of eq 3 with any p or eq 4 with p > 1, the higher-order RDMs can be approximated through their cumulant reconstruction from lower-order RDMs.20 While the present Letter focuses on solving eq 4 with p = 1, the regularization approach developed here is equally applicable to higher-order approximations where p > 1. For p = 1, the excited-state spectrum is computed by solving the following generalized eigenvalue equation for the mth excited-state particle−hole transition amplitude vector cm

Figure 1. Mimic of VF2.1H dye (optical dye).19

Em =

2

p † p ⟨Ψg| Ô Ĥ Ô |Ψg⟩ p

†p

⟨Ψg| Ô Ô |Ψg⟩

Ô =

2

2 i,j Gk , l

in which ci,j(m) are the transition amplitudes to be determined and â†i and âi are the second-quantized operators that create and annihilate an electron in the orbital i, respectively. Evaluation of the excited-state energies from this formula requires knowledge of the ground-state (2p + 2)-RDM. However, we can reduce this dependency to the (2p + 1)RDM Em =

p †p ⟨Ψg| Ô Ô |Ψg⟩

1 + Γ̂ i , j

(8)

=

1 † (aî aĵ + aj†̂ aî ) 2

(9)

From the properties of the generalized eigenvalue equation, the mth and nth excited-state transition amplitudes obey the following orthogonality condition for any m ≠ n

(3)

cTm2Gcn = 0

by using the Schrödinger equation Ĥ |Ψg⟩ = Eg|Ψg⟩. We can further lower the dependency to the 2p-RDM by replacing the p-particle transition operators by their Hermitian analogues

(10)

Because both 2H and 2G are typically singular or nearly singular matrices, the solution of the generalized eigenvalue equation in eq 6 is nontrivial.21 There exist two practical methods to remove the singularities from the generalized eigenvalue problem: (i) the deflation method and (ii) the shift method. The former treats the instabilities by transforming the 2 G matrix into a positive definite matrix in a smaller subspace12 (refer to the Supporting Information). It has been employed in previous work on the extraction of excited states from the ground-state 2-RDM.12 However, the deflation method suffers

(4)

where + p † 1 p Ô = ( Ô + Ô ) 2

1 +1 + = ⟨g | Γ̂ i , j Γ̂ k , l|Ψg⟩

are the elements of the particle−hole RDM, and 1Γ̂+i,j is the Hermitian operator denoting the Hermitian part of the particle−hole transition between orbitals i and j

+ Eg

p + p + 1 ⟨Ψg|[ Ô , [Ĥ , Ô ]]|Ψg⟩ + Eg Em = p +p + 2 ⟨Ψg| Ô Ô |Ψg⟩

1 + 1 + 1 ⟨Ψg|[ Γ̂ i , j, [Ĥ , Γ̂ k , l]]|Ψg⟩ 4 1 + 1 + 1 +1 + 1 + ⟨Ψg|[ Γ̂ k , l , [Ĥ , Γ̂ i , j]]|Ψg⟩ ± Eg ⟨Ψg| Γ̂ i , j Γ̂ k , l|Ψg⟩ 4

are the elements of the effective Hamiltonian matrix

(2)

p † p ⟨Ψg| Ô [Ĥ , Ô ]|Ψg⟩

Hki ,,jl =

(7)

∑ ci(,mj )aî †aĵ i,j

(6)

where

(1)

where the p particle−hole transition operator pÔ accounts for all p transitions including any combination of excitations, deexcitations, and projections relative to the correlated groundstate |Ψg⟩ p

Hcm = E 2Gcm

p

(5) 5374


Letter


(FCI), as well as one- and two-electron integral calculations are performed with the GAMESS electronic structure package.33 The η shift parameter in the Hamiltonian-shifted regularization approach is chosen to be 2.7212 eV. Metallic hydrogen chains with equally spaced hydrogen atoms at a distance of 0.7414 Å occurring under high-pressure conditions were computed with the ES-2RDM method in the double-ζ Dunning−Hay basis set. The vertical band gaps of the hydrogen chains from the 2-RDM method as well as the FCI and CIS methods are shown in Figure 2. The 2-RDM

from instabilities owing to improper choice of cutoff parameter to make the 2G matrix positive semidefinite.12 Here we employ the shift method to remove the singularity from the generalized eigenvalue equation. Consider the transformed generalized eigenvalue equation 2

̃ ̃ = E ̃ 2Gc Hc

(11)

where 2

H̃ = 2 H − α 2G̃

(12)

2

G̃ = 2G̃ − β 2 H̃

(13)

The shift method relies on the fact that the eigenvectors of the generalized eigenvalue problem are invariant to these shift transformations, and hence, the two eigenvalue problems share the same eigenvectors with their eigenvalues related by the following mapping Ẽ =

E−α 1 − βE

(14)

With a suitable choice of the shift parameters, we can remove the singularities of either 2H̃ or 2G̃ or both of these matrices as long as the intersection of their nullspaces is empty. In the present case, a suitable shift is available from choosing α = (Eg − η) and β = 0, where η is a positive number chosen to make the shifted Hamiltonian 2H̃ a positive definite matrix. In the event that the intersection of the 2H̃ and 2G̃ nullspaces is not empty, which has not been observed computationally, the shift method can be combined with deflation. Using a Cholesky decomposition of the shifted Hamiltonian 2H̃ 2

H̃ = LLT

Figure 2. Vertical band gap (eV) for the linear hydrogen chains in the DH basis set using FCI, CIS, and v2-RDM excited-state methods.

method slightly overestimates (∼0.8%) the band gap, while CIS suffers from a relatively large underestimation (∼5%) compared to FCI. It is well-known that CIS and TD-DFT tend to underestimate band gaps due to a lack of explicit electron correlation. Therefore, it is important that the ES-2RDM method, using only single-particle transitions with a correlated ground state, generates accurate band gaps without the overmetallization exhibited by uncorrelated methods. These results as well as results below show that the small gaps in CIS and TD-DFT are due to a deficient or incorrect description of the electron correlation in the ground state rather than limitation of the theory to single-particle excitations. As the length of the hydrogen chain increases, we observe the metalto-insulator transition. Calculations with ground-state 2-RDMs from the v2-RDM and FCI methods, presented in Table S.1, reveal good agreement with the energies differing by less than 0.3% of the excitation energies. Further calculations in this Letter exclusively use ground-state 2-RDMs from the v2-RDM method. Additional data on the hydrogen chains is provided in the Supporting Information. Excited-state energies of n-acene chains (C4n+2H2n+4) were calculated by the ES-2RDM method, CASCI, as well as CIS and TD-DFT. Both the 2-RDM and CASCI methods use an active space of the π orbitals and electrons. For comparison, CASCI employed the natural orbitals from the v2-RDM method’s ground-state 1-RDM. The TD-DFT calculations used the B3LYP DFT functional. To avoid geometric variations with length, we employed geometric parameters from experimental benzene for all chains. Figure 3 displays the vertical band gap for acene chains from our stable ES-v2RDM method. Key conclusions from the data are (i) the errors from the ES-2RDM method are relatively small compared to the uncorrelated single-transition methods, CIS and TD-DFT, (ii)

(15)

we can re-express the generalized eigenvalue problem as an ordinary eigenvalue equation ̃ − T )c ̃ = (L−1 2GL

1 c̃ Ẽ

(16)

in which c̃ = L c. Note that η is chosen to make Ẽ strictly positive, making 1/Ẽ well conditioned. The Hamiltonianshifted regularization not only results in a positive definite transformed Hamiltonian but also avoids the selection of an arbitrary cutoff to remove the numerical zeroes as in the deflation method. Therefore, it is an efficient tool for even the systems with small gaps and low-lying excitations where the deflation method suffers from numerical instabilities. We apply the ES-2RDM method with Hermitian oneparticle transition operators to computing the excited-state spectra of strongly correlated molecules ranging from nickel dithiolates to a mimic of the VF2.1H dye.19 The Hamiltonianshifted regularization yields accurate energies where the traditional deflation method exhibits significant numerical instability. We computed the ground-state 2-RDMs from the v2-RDM method with approximate N-representability conditions known as partial two-positivity constraints.22−32 Except for the calculations on the hydrogen chains, the v2-RDM calculations are performed with an active set of orbitals, that is, a set of orbitals that are correlated. The active-space v2-RDM method (i) minimizes the energy of the active space with the calculation of the 2-RDM from the v2-RDM method, (ii) minimizes the energy with respect to orbital rotations between the active and inactive orbitals, and (iii) repeats steps (i) and (ii) until convergence. TD-DFT, complete-active-space configuration interaction (CASCI), full configuration interaction T

5375


Letter


correlated excited-state energies relative to the CIS and TDDFT methods. Additionally, the 2-RDM method slightly overestimates the band gap relative to the CASCI, in contrast to the CIS and TD-DFT methods, which is consistent with the results from both the hydrogen and acene chains. Table 1 also reveals that the 2-RDM method has fewer missing states than the other methods. Finally, we computed the excited states of a mimic of the VF2.1H dye with the 2-RDM, CASCI, CIS, and TD-DFT methods. The 2-RDM and CASCI approaches used active spaces of 26 electrons in 31 orbitals and 10 electrons in 14 orbitals, respectively. The choice of the active space, which includes both π- and σ-type orbitals, was made by performing an initial v2-RDM calculation.24,26 The ground-state geometry for the mimic was optimized in the abelian point group C2v with DFT using the B3LYP functional in the 6-31G* basis set; the Dunning−Hay double-ζ basis set was employed in subsequent calculations. Table 2 reports the first excited-state

Figure 3. Vertical band gap for the acene chains (eV) in the cc-pVDZ basis set using CASCI, CIS, TD-DFT, and ES-2RDM methods. The CASCI calculations are carried out using only π orbitals.

Table 2. Errors in Excitation Energies (eV) of Different Single-Excitation Methods (CIS, TD-DFT, and v2-RDM Excited-State Methods) for the VF2.1H Dye from Reference 19 with the trans-Stilbene Spacer Removeda

the 2-RDM method slightly (less than 0.4 eV) overestimates the gap, (iii) both CIS and TD-DFT methods underestimate the gap, revealing again overmetallization, and (iv) the band gap decreases with chain length. Detailed results from the acene calculations are presented in the Supporting Information. To study the nickel dithiolate dianion [Ni(edt)2]−2 (see Figure 4), we computed excited states with the ES-2RDM

error ES

S

Deg.

excitation energies CASCI

1 2 3 4 5 6 7

1 0 1 1 0 1 0

3 1 3 3 1 3 1

2.3911 3.4044 3.6353 4.2579 4.4200 4.8465 5.1800

CIS

TD-DFT

ES-v2RDM

−0.6197

−1.2501

−0.7947

−1.6573 −1.9524 −2.0770 −2.4736 −2.6658

0.1659 −0.1658 0.5705 0.3037 0.5574 0.5203 0.4125

−1.1353 −1.5604

a

Energies in the DH basis set are reported relative to those from the CASCI method in a [14,10] active space. The measured band gap for the dye is 2.46 eV, reported in ref 19.

Figure 4. Nickel dithiolates ([Ni(edt)2). The color code is as follows: green, yellow, black, and silver are for Ni, S, C, and H, respectively.

energy relative to the ground-state energy, the optical band gap, as well as higher excited-state energies. The excited-state energies from CASCI and the 2-RDM methods are in good agreement with each other as well as the experimental data from ref 19. Both TD-DFT and CIS methods underestimate the band gap due to the inability of these methods to describe the relatively strong electron correlation in these systems. In this Letter, we present a Hamiltonian-shifted regularization of a family of excited-spectra RDM-based methods for computing excited-state energies from knowledge of groundstate RDMs. In particular, we focus on a specific ES-2RDMbased method with Hermitian single-particle transition operators, also known as the Hermitian operator method.9−16,18 The Hamiltonian-shifted regularization removes the singularities of the generalized eigenvalue equation that are not well-treated by the traditional deflation method. The 2RDM method is explored through its application to a set of strongly correlated molecules including hydrogen and n-acene chains, a nickel dithiolate dianion, and a conjugated dye. Traditional single-excitation methods like TD-DFT and CIS methods suffer from well-documented underestimation of band gaps in both molecules and materials. The ES-2RDM method with its treatment of single excitations from a correlated ground state does not exhibit this problem. In fact, it tends to slightly overestimate the band gap relative to

method, CASCI, CIS, and TD-DFT. Both the 2-RDM and CASCI methods employed an active space of 20 orbitals and 13 electrons [20,13]. As reported by two of the authors in ref 34, the singlet−triplet gap in this molecule is sensitive to the inclusion of dynamic correlation. Table 1 shows the accuracy of the 2-RDM method especially in describing strongly Table 1. Errors in Excitation Energies (eV) of Different Single-Excitation Methods (CIS, TD-DFT, and ES-v2RDM) for Nickel Dithiolates in the cc-pVDZ Basis Set Relative to CASCI[20,13]a error ES

S

Deg.

excitation energies CASCI

1 2 3 4 5 6

0 2 1 0 0 0

1 5 3 1 1 1

3.4967 4.8383 4.8601 4.8709 4.9471 6.0792

CIS

TD-DFT

−2.6722 −4.4625 −4.1820 −4.0192 −4.0056 −4.7648

ES-v2RDM 0.2531 −0.2123 −0.1415 −0.1306 −0.0435 0.7864

a

For each excited state (ES), the spin (S), degeneracy (Deg.), and CASCI energies are reported including only π orbitals. 5376


Letter


(10) Mihailović, M. V.; Rosina, M. The Variational Approach to the Density Matrix for Light Nuclei. Nucl. Phys. A 1975, 237, 221−228. (11) Rosina, M. Application of the Two-Body Density Matrix of the Ground State for Calculations of Some Excited States. Int. J. Quantum Chem. 1978, 13, 737−742. (12) Mazziotti, D. A. Extraction of Electronic Excited States from the Ground-State Two-Particle Reduced Density Matrix. Phys. Rev. A: At., Mol., Opt. Phys. 2003, 68, 052501. (13) Farnum, J. D.; Mazziotti, D. A. Extraction of Ionization Energies from the Ground-State Two-Particle Reduced Density Matrix. Chem. Phys. Lett. 2004, 400, 90−93. (14) Mazziotti, D. A. Quantum Chemistry Without Wave Functions: Two-Electron Reduced Density Matrices. Acc. Chem. Res. 2006, 39, 207−215. (15) Greenman, L.; Mazziotti, A. Electronic Excited-State Energies from a Linear Response Theory Based on the Ground-State TwoElectron Reduced Density Matrix. J. Chem. Phys. 2008, 128, 114109. (16) Valdemoro, C.; Alcoba, D. R.; Oña, O. B.; Tel, L. M.; PérezRomero, E. Combining the G-Particle-Hole Hypervirial Equation and the Hermitian Operator Method to Study Electronic Excitations and De-excitations. J. Math. Chem. 2012, 50, 492−509. (17) van Aggelen, H.; Verstichel, B.; Acke, G.; Degroote, M.; Bultinck, P.; Ayers, P. W.; Van Neck, D. Extended Random Phase Approximation Method for Atomic Excitation Energies from Correlated and Variationally Optimized Second-Order Density Matrices. Comput. Theor. Chem. 2013, 1003, 50−54. (18) Alcoba, D. R.; Massaccesi, G. E.; Oña, O. B.; Torres-Vega, J. J.; Lain, L.; Torre, A. Symmetry-Adapted Formulation of the Combined G-Particle-Hole Hypervirial Equation and Hermitian Operator Method. J. Math. Chem. 2014, 52, 1794−1806. (19) Woodford, C. R.; Frady, E. P.; Smith, R. S.; Morey, B.; Canzi, G.; Palida, S. F.; Araneda, R. C.; Kristan, W. B., Jr; Kubiak, C. P.; Miller, E. W.; et al. Improved PeT Molecules for Optically Sensing Voltage in Neurons. J. Am. Chem. Soc. 2015, 137, 1817−1824. (20) Mazziotti, D. A. Approximate Solution for Electron Correlation through the Use of Schwinger Probes. Chem. Phys. Lett. 1998, 289, 419−427. (21) Saad, Y. Numerical Methods for Large Eigenvalue Problems, revised ed.; Siam, 2011; Vol. 66. (22) Mazziotti, D. A. Variational Minimization of Atomic and Molecular Ground-State Energies via the Two-Particle Reduced Density Matrix. Phys. Rev. A: At., Mol., Opt. Phys. 2002, 65, 062511. (23) Nakata, M.; Nakatsuji, H.; Ehara, M.; Fukuda, M.; Nakata, K.; Fujisawa, K. Variational Calculations of Fermion Second-Order Reduced Density Matrices by Semidefinite Programming Algorithm. J. Chem. Phys. 2001, 114, 8282−8292. (24) Mazziotti, D. A. Realization of Quantum Chemistry without Wave Functions through First-Order Semidefinite Programming. Phys. Rev. Lett. 2004, 93, 213001. (25) Cances, E.; Stoltz, G.; Lewin, M. The Electronic Ground-State Energy Problem: A New Reduced Density Matrix Approach. J. Chem. Phys. 2006, 125, 064101. (26) Shenvi, N.; Izmaylov, A. F. Active-Space N-Representability Constraints for Variational Two-Particle Reduced Density Matrix Calculations. Phys. Rev. Lett. 2010, 105, 213003. (27) Mazziotti, D. A. Large-Scale Semidefinite Programming for Many-Electron Quantum Mechanics. Phys. Rev. Lett. 2011, 106, 083001. (28) Mazziotti, D. A. Enhanced Constraints for Accurate Lower Bounds on Many-Electron Quantum Energies from Variational TwoElectron Reduced Density Matrix Theory. Phys. Rev. Lett. 2016, 117, 153001. (29) Schlimgen, A. W.; Heaps, C. W.; Mazziotti, D. A. Entangled Electrons Foil Synthesis of Elusive Low-Valent Vanadium Oxo Complex. J. Phys. Chem. Lett. 2016, 7, 627−631. (30) DePrince, A. E., III Variational Optimization of the TwoElectron Reduced-Density Matrix under Pure-State N-Representability Conditions. J. Chem. Phys. 2016, 145, 164109.

the CASCI or FCI results. This result demonstrates that band gap underestimation is not caused by restriction to single excitations but rather by the absence or deficiency of electron correlation in the ground state. Because the excited-state 2RDM method only requires the ground-state 2-RDM, it is compatible with any method that can generate a ground-state 2-RDM. Higher excitations can be included through the use of higher RDMs or their approximation by cumulant expansions.12,15 The 2-RDM method has broad potential applications for studying optical band gaps and excitation energies in moderate to strongly correlated molecular systems.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b02455. Further theoretical details as well as additional data from calculations of the hydrogen chains, the acene chains, and the dye (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

S. Hemmatiyan: 0000-0002-0410-6270 D. A. Mazziotti: 0000-0002-9938-3886 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS D.A.M. gratefully acknowledges the United States National Science Foundation Grant CHE-1565638 and the United States Army Research Office (ARO) Grants W911NF-16-C0030 and W911NF-16-1-0152. S.H. and D.A.M. also gratefully acknowledge support from RDMChem LLC.

■

REFERENCES

(1) Okabe, H. Photochemistry of Small Molecules; Wiley Inc., 1978. (2) Falden, H. H.; Falster-Hansen, K. R.; Bak, K. L.; Rettrup, S.; Sauer, S. P. A. Benchmarking Second Order Methods for the Calculation of Vertical Electronic Excitation Energies: Valence and Rydberg states in Polycyclic Aromatic Hydrocarbons. J. Phys. Chem. A 2009, 113, 11995−12012. (3) Snyder, J. W., Jr; Mazziotti, D. A. Photoexcited Conversion of Gauche-1, 3-Butadiene to Bicyclobutane via a Conical Intersection: Energies and Reduced Density Matrices from the Anti-Hermitian Contracted Schrödinger Equation. J. Chem. Phys. 2011, 135, 024107. (4) Foresman, J. B.; Head-Gordon, M.; Pople, J. A.; Frisch, M. J. Toward a Systematic Molecular Orbital Theory for Excited States. J. Phys. Chem. 1992, 96, 135−149. (5) Runge, E.; Gross, E. K. Density-Functional Theory for TimeDependent Systems. Phys. Rev. Lett. 1984, 52, 997. (6) Zangwill, A.; Soven, P. Density-Functional Approach to LocalField Effects in Finite Systems: Photoabsorption in the Rare Gases. Phys. Rev. A: At., Mol., Opt. Phys. 1980, 21, 1561. (7) Casida, M. E. Time-Dependent Density-Functional Theory for Molecules and Molecular Solids. J. Mol. Struct.: THEOCHEM 2009, 914, 3−18. (8) Bartlett, R. J.; Musiał, M. Coupled-Cluster Theory in Quantum Chemistry. Rev. Mod. Phys. 2007, 79, 291. (9) Bouten, M.; Van Leuven, P.; Mihailović, M. V.; Rosina, M. A New Particle-Hole Approach to Collective States. Nucl. Phys. A 1973, 202, 127−144. 5377


Letter

The Journal of Physical Chemistry Letters (31) Sajjan, M.; Mazziotti, D. A. Current-Constrained DensityMatrix Theory to Calculate Molecular Conductivity with Increased Accuracy. Comm. Chem. 2018, 1, 31. (32) Safaei, S.; Mazziotti, D. A. Quantum Signature of Exciton Condensation. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 98, 045122. (33) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; et al. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347−1363. (34) Schlimgen, A. W.; Mazziotti, D. A. Static and Dynamic Electron Correlation in the Ligand Noninnocent Oxidation of Nickel Dithiolates. J. Phys. Chem. A 2017, 121, 9377−9384.

5378


Excited-State Spectra of Strongly Correlated Molecules from a

Recommend Documents