Density functional theory (DFT) based modeling of electronic excited states is of importance for investigation of the photophysical/photochemical properties and spectroscopic characterization of large systems. The widely used linear response time-dependent DFT (TDDFT) approach is however not effective at modeling many types of excited states, including (but not limited to) charge-transfer states, doubly excited states and core-level excitations. In this perspective, we discuss state-specific orbital optimized (OO) DFT approaches as an alterative to TDDFT for electronic excited states. We motivate the use of OO-DFT methods and discuss reasons behind their relatively restricted historical usage (vs TDDFT). We subsequently highlight modern developments that address these factors and allow efficient and reliable OO-DFT computations. Several successful applications of OO-DFT for challenging electronic excitations are also presented, indicating their practical efficacy. OO-DFT approaches are thus increasingly becoming a useful route for computing excited states of large chemical systems. We conclude by discussing the limitations and challenges still facing OO-DFT methods, as well as some potential avenues for addressing them.
We demonstrate that, rather than resorting to high-cost dynamic correlation methods, qualitative failures in excited-state potential energy surface predictions can often be remedied at no additional cost by ensuring that optimal molecular orbitals are used for each individual excited state. This approach also avoids the weighting choices required by state-averaging and dynamic weighting and obviates their need for expensive wave function response calculations when relaxing excited state geometries. Although multi-state approaches are of course preferred near conical intersections, other featur…
Orbital-optimized density functional calculations of excited electronic states: Recent advances and perspectives
Science Institute and Faculty of Physical Sciences, University of Iceland（冰岛大学物理科学学院科学研究所）；Department of Chemical and Pharmaceutical Sciences, University of Trieste（的里雅斯特大学化学与制药科学学院）
A reduced cost equation of motion coupled cluster method for excited states based on state-specific natural orbitals: Theory, Implementation, Benchmark
Localized molecular orbitals are often used for the analysis of chemical bonds, but they can also serve to efficiently and comprehensibly compute linear response properties. While conventional canonical molecular orbitals provide an adequate basis for the treatment of excited states, a chemically meaningful identification of the different excited-state processes is difficult within such a delocalized orbital basis. In this work, starting from an initial set of supermolecular canonical molecular orbitals, we provide a simple one-step top-down embedding procedure for generating a set of orbitals…
Constrained Nuclear-Electronic Orbital Time-Dependent Density Functional Theory for Excited-State Dynamics with Nuclear Quantum Effects
Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison（威斯康星大学麦迪逊分校理论化学研究所和化学系）；Center for Advanced Materials Research, Beijing Normal University（北京师范大学先进材料研究中心）
The Jahn-Teller effect of C$_{60}$ anions in the first electronically excited states was theoretically investigated. The orbital vibronic coupling parameters for the $t_{1g}$ next lowest unoccupied molecular orbitals were derived from the Kohn-Sham orbital levels with hybrid B3LYP functional by using the frozen phonon approach. With the use of these coupling parameters, the vibronic states of the first excited C$_{60}^-$ were derived by exactly diagonalizing the dynamical Jahn-Teller Hamiltonian. The dynamical Jahn-Teller stabilization energy of the first excited C$_{60}^-$ is stronger than th…
Coupled cluster (CC) methods are among the most accurate methods in quantum chemistry. However, the standard CC linear response formulation is not gauge invariant resulting in errors when modelling properties like optical rotation and electron circular dichroism. Including an explicit unitary orbital rotation in the CC Lagrangian makes the linear response function gauge invariant, but the resulting models are not equivalent to full configuration interaction (FCI) in the untruncated limit. In this contribution, such methods are briefly discussed and it is demonstrated that methods using a nonor…
Orbital-optimized Density Functional Calculations of Molecular Rydberg Excited States with Real Space Grid Representation and Self-Interaction Correction
We present a general approach to converge excited state solutions to any quantum chemistry orbital optimization process, without the risk of variational collapse. The resulting Square Gradient Minimization (SGM) approach only requires analytic energy/Lagrangian orbital gradients and merely costs 3 times as much as ground state orbital optimization (per iteration), when implemented via a finite difference approach. SGM is applied to both single determinant $Δ$SCF and spin-purified Restricted Open-Shell Kohn-Sham (ROKS) approaches to study the accuracy of orbital optimized DFT excited states. It…
Herein, we present a new method to efficiently calculate electronically excited states in large molecular assemblies, consisting of hundreds of molecules. For this purpose, we combine the long-range corrected tight-binding density-functional fragment molecular orbital method (FMO-LC-DFTB) with an excitonic Hamiltonian, which is constructed in the basis of locally excited and charge-transfer configuration state functions calculated for embedded monomers and dimers and accounts explicitly for the electronic coupling between all types of excitons. We first evaluate both the accuracy and efficienc…
Measurement Reduction in Orbital-Optimized Variational Quantum Eigensolver via Orbital Compression
State Key Laboratory of Precision and Intelligent Chemistry, University of Science and Technology of China（精密与智能化学国家重点实验室，中国科学技术大学）；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）
The previously proposed ansatz for density cumulant theory that combines orbital-optimization and a parameterization of the 2-electron reduced density matrix cumulant in terms of unitary coupled cluster amplitudes (OUDCT) is carefully examined. Formally, we elucidate the relationship between OUDCT and orbital-optimized unitary coupled cluster theory and show the existence of near-zero denominators in the stationarity conditions for both the exact and some approximate OUDCT methods. We implement methods of the OUDCT ansatz restricted to double excitations for numerical study, up to the fifth co…
The computational investigation of photochemical processes often entails the calculation of excited state geometries, energies, and energy gradients. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons, thereby including the associated nuclear quantum effects and non-Born-Oppenheimer behavior into quantum chemistry calculations. The multicomponent density functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) methods allow efficient calculations of ground and excited states, respectively. Here…
Nuclear-Electronic Orbital Multireference Configuration Interaction for Ground and Excited Vibronic States and Fundamental Insights into Multicomponent Basis Sets
Time-dependent coupled-cluster method with time-varying orbital functions, called time-dependent optimized coupled- cluster (TD-OCC) method, is formulated for multielectron dynamics in an intense laser field. We have successfully derived equations of motion for CC amplitudes and orthonormal orbital functions based on the real action functional, and implemented the method including double excitations (TD-OCCD) and double and triple excitations (TD-OCCDT) within the optimized active orbitals. The present method is size extensive and gauge invariant, a polynomial cost-scaling alternative to the t…
A direct orbital optimization method is presented for density functional calculations of excited electronic states using either a real space grid or a plane wave basis set. The method is variational, provides atomic forces in the excited states, and can be applied to Kohn-Sham (KS) functionals as well as orbital-density dependent functionals (ODD) including explicit self-interaction correction. The implementation for KS functionals involves two nested loops: (1) An inner loop for finding a stationary point in a subspace spanned by the occupied and a few virtual orbitals corresponding to the ex…
We present a comprehensive study of two single-reference approaches to singlet biradicaloids. These two approaches are based on the recently developed regularized orbital-optimized Møller-Plesset method ($κ$-OOMP2). The first approach is to combine the Yamaguchi's approximate projection (AP) scheme and $κ$-OOMP2 with unrestricted (U) orbitals ($κ$-UOOMP2). By capturing only essential symmetry breaking, $κ$-UOOMP2 can serve as a suitable basis for AP. The second approach is $κ$-OOMP2 with complex, restricted (cR) orbitals ($κ$-cROOMP2). Though its applicability is more limited due to the compar…
We propose a quantum-classical hybrid variational algorithm, the quantum orbital minimization method (qOMM), for obtaining the ground state and low-lying excited states of a Hermitian operator. Given parameterized ansatz circuits representing eigenstates, qOMM implements quantum circuits to represent the objective function in the orbital minimization method and adopts classical optimizer to minimize the objective function with respect to parameters in ansatz circuits. The objective function has orthogonality implicitly embedded, which allows qOMM to apply a different ansatz circuit to each ref…
We derive and assess two new classes of regularizers that cope with offending denominators in the single-reference second-order Møller-Plesset perturbation theory (MP2). In particular, we discuss the use of two types of orbital energy dependent regularizers, $κ$ and $σ$, in conjunction with orbital-optimized MP2 (OOMP2). The resulting fifth-order scaling methods, $κ$-OOMP2 and $σ$-OOMP2, have been examined for bond-breaking, thermochemistry, and biradical problems. Both methods with strong enough regularization restore restricted to unrestricted instability (i.e. Coulson-Fischer points) that u…
We applied renormalized singles (RS) in the multireference density functional theory (DFT) to calculate accurate energies of ground and excited states. The multireference DFT approach determines the total energy of the $N$-electron system as the sum of the ($N-2$)-electron energy from a density functional approximation (DFA) and the two-electron addition energies from the particle-particle Tamm-Dancoff approximation (ppTDA), naturally including multireference description. The ppTDA@RS-DFA approach uses the RS Hamiltonian capturing all singles contributions in calculating two-electron addition …
We present the coupled-cluster singles and doubles method formulated in terms of truncated pair-natural orbitals (PNO) that are optimized to minimize the effect of truncation. Compared to the standard ground-state PNO coupled-cluster approaches, in which truncated PNOs derived from first-order Møller-Plesset (MP1) amplitudes are used to compress the CC wave operator, the iteratively-optimized PNOs ("iPNOs") offer moderate improvement for small PNO ranks but rapidly increase their effectiveness for large PNO ranks. The error introduced by PNO truncation in the CCSD energy is reduced by orders o…
Leveraging configuration interaction singles for qualitative descriptions of ground and excited states: state-averaging, linear-response, and spin-projection
College of Engineering, Shibaura Institute of Technology（芝浦工业大学工程学院）；Institute for Molecular Science（分子科学研究所）；Graduate School of Engineering and Science, Shibaura Institute of Technology（芝浦工业大学理工学研究科）
Herein we show that non-resonant inelastic x-ray scattering involving an $s$ core level is a powerful spectroscopic method to characterize the excited states of transition metal compounds. The spherical charge distribution of the $s$ core hole allows the orientational dependence of the intensities of the various spectral features to produce a spatial charge image of the associated multiplet states in a straightforward manner, thereby facilitating the identification of their orbital character. In addition, the $s$ core hole does not add an extra orbital angular momentum component to the multipl…
We show that, as in Hartree Fock theory, the orbitals for excited state mean field theory can be optimized via a self-consistent one-electron equation in which electron-electron repulsion is accounted for through mean field operators. In addition to showing that this excited state ansatz is sufficiently close to a mean field product state to admit a one-electron formulation, this approach brings the orbital optimization speed to within roughly a factor of two of ground state mean field theory. The approach parallels Hartree Fock theory in multiple ways, including the presence of a commutator c…
Perdew-Zunger self-interaction correction (PZ-SIC) offers a route to remove self-interaction errors on an orbital-by-orbital basis. A recent formulation of PZ-SIC by Pederson, Ruzsinszky and Perdew proposes restricting the unitary transformation to localized orbitals called Fermi-Löwdin orbitals. This formulation, called the FLOSIC method, simplifies PZ-SIC calculations and was implemented self-consistently using a Jacobi-like (FLOSIC-Jacobi) iteration scheme. In this work we implement the FLOSIC approach using the Krieger-Li-Iafrate (KLI) approximation to the optimized effective potential (OE…
The pair coupled cluster doubles (pCCD) method (where the excitation manifold is restricted to electron pairs) has a series of interesting features. Among others, it provides ground-state energies very close to what is obtained with doubly-occupied configuration interaction (DOCI), but with polynomial cost (compared with the exponential cost of the latter). Here, we address whether this similarity holds for excited states, by exploring the symmetric dissociation of the linear \ce{H4} molecule. When ground-state Hartree-Fock (HF) orbitals are employed, pCCD and DOCI excited-state energies do no…
We employ a generalized variational principle to improve the stability, reliability, and precision of fully excited-state-specific complete active space self-consistent field theory. Compared to previous approaches that similarly seek to tailor this ansatz's orbitals and configuration interaction expansion for an individual excited state, we find the present approach to be more resistant to root flipping and better at achieving tight convergence to an energy stationary point. Unlike state-averaging, this approach allows orbital shapes to be optimal for individual excited states, which is espec…
The enhancement of the parity-violating energy difference (PVED) by electronic excitation is studied for H$_2X_2$ ($X =$ O, S, Se, Te), CHFClBr, CHFClI, and CHFBrI. To clarify the enhancement mechanism, the dihedral angle dependence of the PVED of H$_2X_2$ in excited states is studied. If the contribution from the highest occupied molecular orbital (HOMO) to the PVED in the ground state is larger than the sum of those from all occupied orbitals, the PVED in the first excited state has a much larger value compared to the ground state due to cancellation breaking among valence orbital contributi…
The variational quantum eigensolver (VQE) algorithm, designed to calculate the energy of molecular ground states on near-term quantum computers, requires specification of symmetries that describe the system, e.g. spin state and number of electrons. This opens the possibility of using VQE to obtain excited states as the lowest energy solutions of a given set of symmetries. In this paper, the performances of various unitary coupled cluster (UCC) ansätze applied to VQE calculations on excited states are investigated, using quantum circuits designed to represent single reference and multireference…
Self-consistent double-hybrid density-functional theory using the optimized-effective-potential method
我们介绍了一个轨道优化的双杂交（DH）计划使用优化有效势（OEP）方法。使用对应于完整的交换相关能量表达式的局部势优化轨道，包括二阶M{\displaystyle M {\math}ller-Plesset（MP2）相关贡献。我们已经实现了一个单参数版本的基于OEP的自洽DH计划使用BLYP密度泛函近似，并将其与相应的非自洽DH计划进行比较，用于计算一些闭壳层原子和分子。虽然基于OEP的自洽性并没有为基态总能量和电离势的计算提供任何改进，但它确实提高了电子亲和势的准确性，并恢复了LUMO轨道能量与中性激发能相关的含义。此外，基于OEP的自洽DH方案提供了相当准确的交换相关势和相关密度。
We investigate an extension of excited state mean-field theory in which the energy expression is augmented with density functional components in an effort to include the effects of weak electron correlations. The approach remains variational and entirely time-independent, allowing it to avoid some of the difficulties associated with linear response and the adiabatic approximation. In particular, all of the electrons' orbitals are relaxed state specifically and there is no reliance on Kohn-Sham orbital energy differences, both of which are important features in the context of charge transfer. P…
