Very accurate potential energy curve of the LiH molecule

Molecular Structure Theory without the Born-Oppenheimer Approximation: Rotationless Vibrational States of LiH

Low-lying rotationless states of the lithium hydride molecule are studied in the framework of the variational method without assuming the Born-Oppenheimer (BO) approximation. Highly accurate solutions to the six-particle (two nuclei and four electrons) Schrödinger equation are obtained by means of expanding the wave functions of the considered states in terms of many thousands of all-particle explicitly correlated Gausssians. The basis functions are optimized independently for each state using the analytic energy gradient with respect to the nonlinear parameters. The non-BO wave functions obtained in the calculations are used to evaluate the leading-order relativistic and quantum electrodynamics energy corrections in the framework of the perturbation theory. The geometric structure of the molecule in the ground and excited states is discussed based on the analysis of the nucleus-nucleus correlation functions. The non-BO energies and structural parameters obtained of this work are compared with the most accurate BO results currently available.

Subspace-Search Quantum Imaginary Time Evolution for Excited State Computations

Quantum systems in excited states are attracting significant interest with the advent of noisy intermediate-scale quantum (NISQ) devices. While ground states of small molecular systems are typically explored using hybrid variational algorithms like the variational quantum eigensolver (VQE), the study of excited states has received much less attention, partly due to the absence of efficient algorithms. In this work, we introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices by integrating key elements of the subspace search variational quantum eigensolver (SSVQE) and the variational quantum imaginary time evolution (VarQITE) method. The effectiveness of SSQITE is demonstrated through calculations of low-lying excited states of benchmark model systems including H2 and LiH molecules. A toy Hamiltonian is also employed to demonstrate that the robustness of VarQITE in avoiding local minima extends to its use in excited state algorithms. With this robustness in avoiding local minima, SSQITE shows promise for advancing quantum computations of excited states across a wide range of applications.

Analytic Nuclear Gradients for Complete Active Space Linearized Pair-Density Functional Theory

Accurately modeling photochemical reactions is difficult due to the presence of conical intersections and locally avoided crossings, as well as the inherently multiconfigurational character of excited states. As such, one needs a multistate method that incorporates state interaction in order to accurately model the potential energy surface at all nuclear coordinates. The recently developed linearized pair-density functional theory (L-PDFT) is a multistate extension of multiconfiguration PDFT, and it has been shown to be a cost-effective post-MCSCF method (as compared to more traditional and expensive multireference many-body perturbation methods or multireference configuration interaction methods) that can accurately model potential energy surfaces in regions of strong nuclear-electronic coupling in addition to accurately predicting Franck-Condon vertical excitations. In this paper, we report the derivation of analytic gradients for L-PDFT and their implementation in the PySCF-forge software, and we illustrate the utility of these gradients for predicting ground- and excited-state equilibrium geometries and adiabatic excitation energies for formaldehyde, s-trans-butadiene, phenol, and cytosine.

Deformed explicitly correlated Gaussians

Deformed explicitly correlated Gaussian (DECG) basis functions are introduced, and their matrix elements are calculated. All matrix elements can be calculated analytically in a closed form, except the Coulomb one, which has to be approximated by a Gaussian expansion. The DECG basis functions can be used to solve problems with nonspherical potentials. One example of such potential is the dipole self-interaction term in the Pauli-Fierz Hamiltonian. Examples are presented showing the accuracy and necessity of deformed Gaussian basis functions to accurately solve light-matter coupled systems in cavity QED.

Accurate spectroscopic properties by diffusion quantum Monte Carlo calculations

The capability of Diffusion Quantum Monte Carlo (DMC) to produce high quality potential energy curve (PEC) was evaluated. H₂⁺, HeH⁺ and LiH PECs were built by all-electron fixed-node DMC calculations. Trial wave functions were obtained from Hartree-Fock (HF) (H₂⁺), MCSCF and CI (HeH⁺ and LiH) calculations multiplied by Jastrow factor. The quality of these generated PECs was analyzed throughout equilibrium distance, dissociation energy, vibrational energies and rovibrational spectroscopic constants (ω_e, ω_ex_e, ω_ey_e, α_e, γ_e and B_e). The Discrete Variable Representation (DVR) and the Dunham approaches were used to determine the rovibrational spectroscopic constants. The PECs and the aforementioned properties were also obtained by the following methods: MCSCF/aug-cc-pV5Z (LiH), CCSD(T)/aug-cc-pV5Z (HeH⁺ and LiH) and HF (H₂⁺ and HeH⁺) levels. The results of these DMC computations, specially the DMC-DVR procedure, are the most accurate among others DMC calculations available in the literature for these systems. They suggest that DMC can be used to achieve accurate PECs to produce spectroscopic properties with the same level of accuracy of theoretical benchmarks and experimental data of the literature.

Compressed-State Multistate Pair-Density Functional Theory

Multiconfiguration pair-density functional theory (MC-PDFT) is a multireference method that can be used to calculate excited states. However, MC-PDFT potential energy surfaces have the wrong topology at conical intersections because the last step of MC-PDFT is not a diagonalization of a model-space Hamiltonian matrix, as done in, for example, multistate second-order perturbation theory (MS-CASPT2). We have previously proposed methods that solve this problem by diagonalizing a model-space effective Hamiltonian matrix, where the diagonal elements are MC-PDFT energies for intermediate states, and the off-diagonal elements are evaluated by wave function theory. One previous method is called variational multistate PDFT (VMS-PDFT), whose intermediate states maximize the trace of the effective Hamiltonian, namely, the sum of the MC-PDFT energies of the model-space states; the VMS-PDFT is very robust but is more computationally expensive than another method, extended multistate PDFT (XMS-PDFT), in which the transformation to intermediate states is accomplished without needing any density functional evaluations. However, although VMS-PDFT was accurate in all cases tested, XMS-PDFT was accurate in only some of them. In the present paper, we propose a new method, called compressed-state multistate PDFT (CMS-PDFT), that is as efficient as XMS-PDFT and as accurate as VMS-PDFT. The new method maximizes the trace of the classical Coulomb energy of the intermediate states such that the electron densities of the intermediate states are compressed. We show that CMS-PDFT performs robustly even where XMS-PDFT fails.

Singlet-triplet gaps in diradicals obtained with diffusion quantum Monte Carlo using a Slater-Jastrow trial wavefunction with a minimum number of determinants

Diradicals are essential species in a wide range of chemical processes, whereas the computational study of their electronic structure often remains a challenge due to near-degeneracy of the frontier molecular orbitals. The fixed-node diffusion quantum Monte Carlo (FN-DMC) method is employed to calculate adiabatic energy gaps of some typical diradicals with the Slater-Jastrow trial wavefunction. The antisymmetrized part of the trial wavefunction is taken to be a linear combination of a minimum number of determinants using RB3LYP orbitals from the closed-shell singlet state or ROB3LYP orbitals from the triplet state. Our results show that using the two-determinant-Jastrow trial wavefunction is necessary to achieve reliable energy differences between closed-shell singlet states. The energy of the triplet state with MS = 1 is calculated to be lower than that with MS = 0 with FN-DMC even using trial wavefunctions with spin-pure states as their antisymmetrized parts and this difference is reduced with better orbitals. This indicates that the fixed-node error is smaller for the triplet state with MS = 1. Adiabatic energy gaps obtained from the present FN-DMC calculations are in reasonable agreement with available experimental values. Compared with results of the high level EOM-SF-CC method, energy gaps of FN-DMC with RB3LYP orbitals are slightly better than those using ROB3LYP orbitals and results of EOM-SF-CCSD. The present FN-DMC calculations using the simplest ansatz for the trial wavefunction can achieve reasonable results for these diradicals and they can readily be applied to large diradicals.

Solving the Schrödinger equation of atoms and molecules with the free-complement chemical-formula theory: First-row atoms and small molecules

The free-complement chemical-formula theory (FC-CFT) for solving the Schrödinger equation (SE) was applied to the first-row atoms and several small molecules, limiting only to the ground state of a spin symmetry. Highly accurate results, satisfying chemical accuracy (kcal/mol accuracy for the absolute total energy), were obtained for all the cases. The local Schrödinger equation (LSE) method was applied for obtaining the solutions accurately and stably. For adapting the sampling method to quantum mechanical calculations, we developed a combined method of local sampling and Metropolis sampling. We also reported the method that leads the calculations to the accurate energies and wave functions as definite converged results with minimum ambiguities. We have also examined the possibility of the stationarity principle in the sampling method: it certainly works, though more extensive applications are necessary. From the high accuracy and the constant stability of the results, the present methodology seems to provide a useful tool for solving the SE of atoms and molecules.

How well can density functional theory and pair-density functional theory predict the correct atomic charges for dissociation and accurate dissociation energetics of ionic bonds?

The accuracy of density functional theory (DFT) is often judged by predicted dissociation energies, but one should also consider charge densities as illustrated here for dissociation of heteronuclear diatomic molecules, including ionic bonds for which local density functionals yield erroneous results. Some hybrid density functionals with 100% exact exchange in Kohn-Sham DFT and the local functionals in multiconfiguration pair-density functional theory give relatively acurate dissociation energies for NaCl, and they correctly yield uncharged dissociated atoms.

Electronic structure and time-dependent description of rotational predissociation of LiH

The adiabatic potential energy curves of the ¹Σ⁺ and ¹Π states of the LiH molecule were calculated. They correlate asymptotically to atomic states, such as 2s + 1s, 2p + 1s, 3s + 1s, 3p + 1s, 3d + 1s, 4s + 1s, 4p + 1s and 4d + 1s. A very good agreement was found between our calculated spectroscopic parameters and the experimental ones. The dynamics of the rotational predissociation process of the 1¹Π state were studied by solving the time-dependent Schrödinger equation. The classical experiment of Velasco [Can. J. Phys., 1957, 35, 1204] on dissociation in the 1¹Π state is explained for the first time in detail.

Two-photon absorption of the spatially confined LiH molecule

In the present contribution we study the influence of spatial restriction on the two-photon dipole transitions between the X¹Σ⁺ and A¹Σ⁺ states of lithium hydride. The bond-length dependence of the two-photon absorption strength is also analyzed for the first time in the literature. The highly accurate multiconfiguration self-consistent field (MCSCF) method and response theory are used to characterize the electronic structure of the studied molecule. In order to render the effect of orbital compression we apply a two-dimensional harmonic oscillator potential, mimicking the topology of cylindrical confining environments (e.g. carbon nanotubes, quantum wires). Among others, the obtained results provide evidence that at large internuclear distances the TPA response of lithium hydride may be significantly enhanced and this effect is much more pronounced upon embedding of the LiH molecule in an external confining potential. To understand the origin of the observed variation in the two-photon absorption response a two-level approximation is employed.

A simplified ab initio treatment of diradicaloid structures produced from stretching and breaking chemical bonds

With a proper choice of active spaces, the single root perturbation theory employing improved virtual orbitals can flawlessly describe the ground, excited, ionized, and dissociated states having varying degrees of degeneracy at the expense of low computational cost.

Orbit-orbit relativistic correction calculated with all-electron molecular explicitly correlated Gaussians

An algorithm for calculating the first-order electronic orbit-orbit magnetic interaction correction for an electronic wave function expanded in terms of all-electron explicitly correlated molecular Gaussian (ECG) functions with shifted centers is derived and implemented. The algorithm is tested in calculations concerning the H₂ molecule. It is also applied in calculations for LiH and H₃⁺ molecular systems. The implementation completes our work on the leading relativistic correction for ECGs and paves the way for very accurate ECG calculations of ground and excited potential energy surfaces (PESs) of small molecules with two and more nuclei and two and more electrons, such as HeH^-, H₃⁺, HeH₂⁺, and LiH₂⁺. The PESs will be used to determine rovibrational spectra of the systems.

Algorithms for calculating mass-velocity and Darwin relativistic corrections with n-electron explicitly correlated Gaussians with shifted centers

Algorithms for calculating the leading mass-velocity (MV) and Darwin (D) relativistic corrections are derived for electronic wave functions expanded in terms of n-electron explicitly correlated Gaussian functions with shifted centers and without pre-exponential angular factors. The algorithms are implemented and tested in calculations of MV and D corrections for several points on the ground-state potential energy curves of the H2 and LiH molecules. The algorithms are general and can be applied in calculations of systems with an arbitrary number of electrons.

Description of Polar Chemical Bonds from the Quantum Mechanical Interference Perspective

The Generalized Product Function Energy Partitioning (GPF-EP) method has been applied to a set of molecules, AH (A = Li, Be, B, C, N, O, F), CO and LiF with quite different dipole moments, in order to investigate the role played by the quantum interference effect in the formation of polar chemical bonds. The calculations were carried out with GPF wave functions treating all the core electrons as a single Hartree-Fock group and the bonding electrons at the Generalized Valence Bond Perfect-Pairing (GVB-PP) level, with the cc-pVTZ basis set. The results of the energy partitioning into interference and quasi-classical contributions along the respective Potential Energy Surfaces (PES) show that the main contribution to the depth of the potential wells comes from the interference term, which is an indication that all the molecules mentioned above form typical covalent bonds. In all cases, the stabilization promoted by the interference term comes from the kinetic contribution, in agreement with previous results. The analysis of the effect of quantum interference on the electron density reveals that while polarization effects (quasi-classical) tend to displace electronic density from the most polarizable atom toward the less polarizable one, interference (quantum effects) counteracts by displacing electronic density to the bond region, giving rise to the right electronic density and dipole moment.

Studies of singlet Rydberg series of LiH derived from Li(nl) + H(1s), with n ≤ 6 and l ≤ 4

The 50 singlet states of LiH composed of 49 Rydberg states and one non-Rydberg ionic state derivable from Li(nl) + H(1s), with n ≤ 6 and l ≤ 4, are studied using the multi-reference configuration interaction method combined with the Stuttgart/Köln group's effective core potential/core polarization potential method. Basis functions that can yield energy levels up to the 6g orbital of Li have been developed, and they are used with a huge number of universal Kaufmann basis functions for Rydberg states. The systematics and regularities of the physical properties such as potential energies, quantum defects, permanent dipole moments, transition dipole moments, and nonadiabatic coupling matrix elements of the Rydberg series are studied. The behaviors of potential energy curves and quantum defect curves are explained using the Fermi approximation. The permanent dipole moments of the Rydberg series reveal that they are determined by the sizes of the Rydberg orbitals, which are proportional to n(2). Interesting mirror relationships of the dipole moments are observed between l-mixed Rydberg series, with the rule Δl = ±1, except for s-d mixing, which is also accompanied by n-mixing. The members of the l-mixed Rydberg series have dipole moments with opposite directions. The first derivatives of the dipole moment curves, which show the charge-transfer component, clearly show not only mirror relationships in terms of direction but also oscillations. The transition dipole moment matrix elements of the Rydberg series are determined by the small-r region, with two consequences. One is that the transition dipole moment matrix elements show n(-3/2) dependence. The other is that the magnitudes of the transition dipole moment matrix elements decrease rapidly as l increases.

Analytical energy gradient used in variational Born-Oppenheimer calculations with all-electron explicitly correlated Gaussian functions for molecules containing one π electron

An algorithm for variational calculations of molecules with one π electron performed with all-electron explicitly correlated gaussian (ECG) functions with floating centers is derived and implemented. The algorithm includes the analytic gradient of the Born-Oppenheimer electronic energy determined with respect to the ECG exponential parameters and the coordinates of the gaussian centers. The availability of the gradient greatly accelerates the variational energy minimization. The algorithm is tested in calculations of four electronic excited states, c(3)Π(u), C(1)Π(u), i(3)Π(g), and I(1)Π(g), of the hydrogen molecule at a single internuclear distance specific to each state. With the use of the analytical energy gradient, the present calculations yield new, lowest-to-date, variational energy upper bounds for all four states.

Electric dipole (hyper)polarizabilities of spatially confined LiH molecule

In this study we report on the electronic contributions to the linear and nonlinear static electronic electric dipole properties, namely the dipole moment (μ), the polarizability (α), and the first-hyperpolarizability (β), of spatially confined LiH molecule in its ground X (1)Σ(+) state. The finite-field technique is applied to estimate the corresponding energy and dipole moment derivatives with respect to external electric field. Various forms of confining potential, of either spherical or cylindrical symmetry, are included in the Hamiltonian in the form of one-electron operator. The computations are performed at several levels of approximation including the coupled-cluster methods as well as multi-configurational (full configuration interaction) and explicitly correlated Gaussian wavefunctions. The performance of Kohn-Sham density functional theory for the selected exchange-correlation functionals is also discussed. In general, the orbital compression effects lead to a substantial reduction in all the studied properties regardless of the symmetry of confining potential, however, the rate of this reduction varies depending on the type of applied potential. Only in the case of dipole moment under a cylindrical confinement a gradual increase of its magnitude is observed.

Very accurate potential energy curve of the He2+ ion

A very accurate ground-state potential energy curve (PEC) of the He(2)(+) molecule is calculated with 1200 explicitly correlated Gaussian functions with shifted centers in the range between 0.9 and 100 a(0). The calculations include the adiabatic corrections determined for the (3)He(4)He(+), (3)He(2)(+), and (4)He(2)(+) isotopologues. The absolute accuracy of the PEC is better than 0.05 cm(-1) and that of the adiabatic corrections is around 0.01 cm(-1). The depths of the PECs augmented with the adiabatic corrections for the three isotopologues are: 19 956.708 cm(-1) for (4)He(2)(+), 19 957.054 cm(-1) for (3)He(4)He(+), and 19 957.401 cm(-1) for (3)He(2)(+). The rovibrational energies are also determined. For (3)He(4)He(+) the computed rovibrational transitions corresponding to the ν = 1-0 band differ from the experiment by less than 0.005 cm(-1). For the rovibrational transitions corresponding to the ν = 23-22 band the difference is around 0.012 cm(-1). Presently, this represents the best agreement between theory and experiment for He(2)(+).