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 reference state. We carry out numerical simulations that seek to find excited states of the $\text{H}_{2}$, $\text{LiH}$, and a toy model consisting of 4 hydrogen atoms arranged in a square lattice in the STO-3G basis and UCCSD ansatz circuits. Comparing the numerical results with existing excited states methods, qOMM is less prone to getting stuck in local minima and can achieve convergence with more shallow ansatz circuits.
It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nod…
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…
Correct Small-Truncated Excited State Wave functions Obtained via Minimization Principle for Excited States compared / opposed to Hylleraas-Undheim and McDonald higher roots
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…
A reduced cost equation of motion coupled cluster method for excited states based on state-specific natural orbitals: Theory, Implementation, Benchmark
We investigate the issue of optimization stability in variance-based state-specific variational Monte Carlo, discussing the roles of the objective function, the complexity of wave function ansatz, the amount of sampling effort, and the choice of minimization algorithm. Using a small cyanine dye molecule as a test case, we systematically perform minimizations using variants of the linear method as both a standalone algorithm and in a hybrid combination with accelerated descent. We demonstrate that adaptive step control is crucial for maintaining the linear method's stability when optimizing com…
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 motiv…
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）
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…
Orbital-optimized Density Functional Calculations of Molecular Rydberg Excited States with Real Space Grid Representation and Self-Interaction Correction
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…
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…
In the Noisy Intermediate-Scale Quantum (NISQ) era, solving the electronic structure problem from chemistry is considered as the "killer application" for near-term quantum devices. In spite of the success of variational hybrid quantum/classical algorithms in providing accurate energy profiles for small molecules, careful considerations are still required for the description of complicated features of potential energy surfaces. Because the current quantum resources are very limited, it is common to focus on a restricted part of the Hilbert space (determined by the set of active orbitals). While…
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 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
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…
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 perform excited-state variational Monte Carlo and diffusion Monte Carlo calculations using a simple and efficient wave function ansatz. This ansatz follows the recent variation-after-response formalism, accurately approximating a configuration interaction singles wave function as a sum of only two non-orthogonal Slater determinants, and further including important orbital relaxation. The ansatz is used to perform diffusion Monte Carlo calculations with large augmented basis sets, comparing to benchmarks from near-exact quantum chemical methods. The significance of orbital optimization in ex…
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）
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…
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…
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…
Variational calculations of excited electronic states are carried out by finding saddle points on the surface that describes how the energy of the system varies as a function of the electronic degrees of freedom. This approach has several advantages over commonly used methods especially in the context of density functional calculations, as collapse to the ground state is avoided and yet, the orbitals are variationally optimized for the excited state. This optimization makes it possible to describe excitations with large charge transfer where calculations based on ground state orbitals are prob…
The energy minimization involved in density functional calculations of electronic systems can be carried out using an exponential transformation that preserves the orthonormality of the orbitals. The energy of the system is then represented as a function of the elements of a skew-Hermitian matrix that can be optimized directly using unconstrained minimization methods. An implementation based on the limited memory Broyden-Fletcher-Goldfarb-Shanno approach with inexact line search and a preconditioner is presented and the performance compared with that of the commonly used self-consistent field …
We present a discussion of recent progress in excited-state-specific quantum chemistry and quantum Monte Carlo alongside a demonstration of how a combination of methods from these two fields can offer reliably accurate excited state predictions across singly excited, doubly excited, and charge transfer states. Both of these fields have seen important advances supporting excited state simulation in recent years, including the introduction of more effective excited-state-specific optimization methods, improved handling of complicated wave function forms, and ways of explicitly balancing the qual…
State-specific orbital optimized approaches are more accurate at predicting core-level spectra than traditional linear-response protocols, but their utility had been restricted on account of the risk of `variational collapse' down to the ground state. We employ the recently developed square gradient minimization (SGM, J. Chem. Theory Comput. 16, 1699-1710, 2020) algorithm to reliably avoid variational collapse and study the effectiveness of orbital optimized density functional theory (DFT) at predicting second period element 1s core-level spectra of open-shell systems. Several density function…
This paper reports an implementation of Hartree-Fock linear response with complex orbitals for computing electronic spectra of molecules in a strong external magnetic fields. The implementation is completely general, allowing for spin-restricted, spin-unrestricted, and general two-component reference states. The method is applied to small molecules placed in strong uniform and non-uniform magnetic fields of astrochemical importance at the Random Phase Approximation level of theory. For uniform fields, where comparison is possible, the spectra are found to be qualitatively similar to those rece…
