Potential energy surfaces and properties of the Br–HBr complex

Simulations of electronically nonadiabatic processes may employ either the adiabatic or diabatic representation. Direct dynamics calculations are usually carried out in the adiabatic basis because the energy, force, and state coupling can be evaluated directly by many electronic structure methods. However, although its straightforwardness is appealing, direct dynamics is expensive when combined with quantitatively accurate electronic structure theories. This generates interest in analytically fitted surfaces to cut the expense, but the cuspidal ridges of the potentials and the singularities and vector nature of the couplings at high-dimensional, nonsymmetry-determined intersections in the adiabatic representation make accurate fitting almost impossible. This motivates using diabatic representations, where the surfaces are smooth and the couplings are also smooth and-importantly-scalar. In a recent previous work, we have developed a method called diabatization by deep neural network (DDNN) that takes advantage of the smoothness and nonuniqueness of diabatic bases to obtain them by machine learning. The diabatic potential energy matrices (DPEMs) learned by the DDNN method yield not only diabatic potential energy surfaces (PESs) and couplings in an analytic form useful for dynamics calculations, but also adiabatic surfaces and couplings in the adiabatic representation can be calculated inexpensively from the transformation. In the present work, we show how to extend the DDNN method to produce good approximations to global permutationally invariant adiabatic PESs simultaneously with DPEMs. The extended method is called permutationally restrained DDNN.

Diabatic and adiabatic representations: Electronic structure caveats

In this paper, we describe implementation of the semiclassical Liouville method for simulating molecular dynamics on coupled electronic surfaces in the electronic adiabatic representation. We cast the formalism in terms of semiclassical motion on Born–Oppenheimer potential energy surfaces with nonadiabatic coupling arising from the coordinate dependence of the adiabatic electronic eigenstates. Using perturbation theory and asymptotic evaluation of the resulting time integrals, we derive an expression for the probability of transition between adiabatic states which agrees with the result given previously by Miller and George [W. H. Miller and T. F. George, J. Chem. Phys. 56, 5637 (1972)]. We also demonstrate numerically the equivalence of semiclassical trajectory-based calculations in the adiabatic and diabatic representations by performing molecular dynamics simulations on a model two-state system and comparing with exact quantum mechanical results. Excellent agreement between the exact and semiclassical treatments is obtained in both representations.

A method for fitting ab initio determined spin-orbit coupling interactions, in the Breit-Pauli approximation, based on quasidiabatic representations using neural network fits is reported. The algorithm generalizes our recently reported neural network approach for representing the dipole interaction. The S0, S1, and T1 states of formaldehyde are used as an example. First, the two singlet states S0 and S1 are diabatized with a modified Boys Localization diabatization method. Second, the spin-orbit coupling between singlet and triplet states is transformed to the diabatic representation. This removes the discontinuities in the adiabatic representation. The diabatized spin-orbit couplings are then fit with smooth neural network functions. The analytic representation of spin-orbit coupling interactions in a diabatic basis by neural networks will make accurate full-dimensional quantum dynamical treatment of both internal conversion and intersystem crossing possible, which will help us to gain better understanding of both processes.

The Born-Oppenheimer approximation assumes nuclear motion evolves on single, uncoupled potential energy surfaces, widely used to solve the time-independent Schrödinger equation for atomistic systems. However, for near-degenerate same-symmetry electronic states, avoided crossings in the potential energy curves occur and non-adiabatic couplings (NACs) become significant. In such cases, the adiabatic approximation is unsuitable for high-resolution spectroscopy. A unitary transformation to the diabatic representation can eliminate NACs, resulting in smooth molecular property curves that may cross. Computing this adiabatic-to-diabatic transformation (AtDT) is desirable but non-analytic for multi-state coupled systems, necessitating numerical solutions. It remains unclear if current methods yield numerically exact AtDTs ensuring rovibronic energy level equivalence between adiabatic and diabatic pictures. We demonstrate (for the first time) numerically exact equivalence of adiabatic and diabatic representations for -state diatomic molecules using abinitio data for , CH, and a model 10-state system. We show how the equivalence can be efficiently used to assess the importance of non-adiabatic effects and the impact of omitting them when computing rovibronic energies of diatomic molecules. The adiabatic and diabatic representations of the spectroscopic model, including all coupling terms, have been implemented in the diatomic code Duo.

An analytical polynomial formulation of the exponential of fully symmetric matrices is proposed. This polynomial formula is derived using the Cayley - Hamilton theorem and the polynomial Euclidean division. The application field of these developments is time-dependent wavepacket calculations involving multiple potential energy surface crossing. The use of this polynomial formulation in time-dependent calculations is directly connected to the choice of the split operator, proposed by Feit and Fleck (1982 J. Comput. Phys. 47 412) as a numerical propagation scheme. We show in this paper that this analytical polynomial formula is numerically more efficient than the scheme proposed by Almeida and Metiu (1988 Chem. Phys. Lett. 146 47) where the exponential of the kinetic operator is evaluated in the momentum and diabatic representation and the exponential of the potential operator is evaluated in the coordinate and adiabatic representation. The use of an optical potential ( negative imaginary potential) to absorb the wavepacket near the edges of the grid point is quite simple to implement as it appears as a multiplicative factor in front of the polynomial expression. Our scheme is exemplified on the photodissociation of a diatomic molecule involving three potential energy curves.

Part I presents several sets of comparisons of semi-classical, quasi-classical and exact quantum reactive scattering calculations for collinear chemical reactions. The possibility of modifying the standard quasi-classical method according to a quantum criterion is investigated. The systems studied are H + H_2, F + H_2, and F + D_2. In addition, a theoretical investigation of the semi-classical S matrix is made. Details of a quasi-classical current density analysis of the H + H_2 reaction are presented and a comparison with exact quantum results is made. A direct test of two versions of the vibrationally adiabatic theory of chemical reactions is made in Part II for the H + H_2 reaction. The adiabaticity of the symmetric stretch motion of the H_3 transition state is focussed upon. In addition, a determination of the completeness of adiabatic basis sets for scattering calculations is made. The theory of electronically non-adiabatic chemical reactions is presented in Part III. Quantum calculations of the collinear H^+ + H_2 → H_2 + H^+ reaction are described. A model and a realistic potential energy surface are employed in these calculations. A fictitious electronically non-adiabatic H + H_2 collinear chemical reaction is treated quantum mechanically. Two potential energy surfaces and a coupling surface are developed for this purpose. The reaction Ba(^1S) + ON_2(X^1Σ) → BaO(X^1Σ) + N_2(X^1Σ^+_g), BaO(a^3II) + N_2(X^1Σ^+_g) is studied quantum mechanically. The singlet and triplet potential energy surfaces are devised as is a spin-orbit coupling surface. Electronically adiabatic and non-adiabatic transition probabilities are calculated as a function of the initial translational energy of the reagents.

Full-potential all-electron density-functional calculations with mixed basis augmented plane wave plus local orbital--linearized-augmented plane wave have been carried out to investigate the electronic and geometric properties of bulk FCC Am and quantum size effects in the surface energies and the work functions of the (111) ultrathin films up to seven layers at six levels of approximations, namely, non-spin-polarized--no-spin-orbit coupling, non-spin-polarized--spin-orbit coupling, spin-polarized--no-spin-orbit coupling, spin-polarized--spin-orbit coupling, antiferromagnetic--no-spin-orbit coupling, and antiferromagnetic--spin-orbit (AFM-SO) coupling. The antiferromagnetic state including spin-orbit coupling is found to be the ground state of FCC Am with the $5f$ electrons primarily localized. Our calculations show that both spin-polarization and spin-orbit coupling effects play important roles in determining the equilibrium lattice constant, the bulk modulus, as well as the localized feature of $5f$ electrons for FCC Am. Our calculated equilibrium lattice constant of $9.21\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and a bulk modulus of $25.64\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ at the AFM-SO level are in good agreement with the experimental values of $9.26\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and $29.4\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, respectively. For the films, surface energy rapidly converges and the semi-infinite surface energy is predicted to be $0.81\phantom{\rule{0.3em}{0ex}}\mathrm{J}∕{\mathrm{m}}^{2}$ at the ground state. The work function of the FCC Am(111) surface at the AFM-SO ground state is predicted to be $3.06\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, and the work function shows some oscillations when the number of layers is less than five, while it becomes relatively stable when the number of layers is greater than five. This suggests that a five-layer film may be required for any future adsorption investigation that requires, for example, an accurate prediction of one-electron properties and adsorbate-induced work function shift.

A simple procedure is described for calculating spin-orbit and nonadiabatic interactions by an extension of the diatomics-in-molecules approach. The method can be easily applied to a large number of polyatomic systems and will hopefully be of value in obtaining qualitative and quantitative descriptions of many important molecular collision processes which involve more than one potential energy surface. Results are presented for two such processes, the quenching of Li(2 2P) by H2 and the reaction of F atoms with H2.

Chapter 2 is a tutorial on computing spin-forbidden reactions. Most chemists solving problems with quantum chemical tools typically work on a single potential energy surface but there are many chemical transformations where two or more potential energy surfaces need to be included to describe properly the event that is taking place. How one goes about treating changes in spin multiplicity in this situation is a daunting task. Described for the novice is the importance of the minimum energy crossing point (MEXP) and a rationalization of how spin–orbit coupling provides a mechanism for spin-forbidden reactions. An explanation of crossing probabilities, the Fermi golden rule and the Landau–Zener semiclassical approximation are given. Methodologies for obtaining spin–orbit matrix elements are presented including, among others, the Klein–Gordon equation, the Dirac equation, the Foldy–Wouthuysen transformation and the Breit–Pauli Hamiltonian. With this background the novice is taken through a tutorial that explains how to locate the MEXP. Programs available for modeling spin-forbidden reactions are provided and examples of such calculations on diatomic and polyatomic molecules are given. The chapter covers: Overview of Reactions Requiring Two States Spin-Forbidden Reaction, Intersystem Crossing Spin–Orbit Coupling as a Mechanism for Spin-Forbidden Reactions General Considerations Atomic Spin–Orbit Coupling Molecular Spin–Orbit Coupling Crossing Probability Fermi Golden Rule Landau–Zener Semiclassical Approximation Methodology for Obtaining Spin–Orbit Matrix Elements Electron Spin in Non-Relativistic Quantum Mechanics Klein–Gordon Equation Dirac Equation Foldy–Wouthuysen Transformation Breit–Pauli Hamiltonian Zeff Method Effective Core Potential-Based Method Model Core Potential-Based Method Douglas–Kroll Transformation Potential Energy Surfaces Minimum Energy Crossing Point Location Available Programs for Modeling Spin-Forbidden Reactions Applications to Spin-Forbidden Reactions Diatomic Molecules Polyatomic Molecules Molecular Properties Dynamical Aspects Other Reactions Biological Chemistry

As a kind of toxic methylation reagent and disinfectant, methyl iodide (CH3I) is widely used. It is important to study the basic physical properties of methyl iodide and to use effective measures to degrade it. The ground states of methyl iodide in different electric fields from 0 to 0.04a.u.(atomic unit) are optimized using the B3LYP calculation with the LANL2DZ basis set. Optimized parameters, total energies, bond lengths, dipole moments, the highest occupied molecular orbital energies, the lowest unoccupied molecular orbital energies, energy gaps, infrared spectra, and dissociation potential energy surface (PES) are obtained. The obtained results show that when the external electric field gradually increases from 0 to 0.04a.u. along the molecular axis Z (the C-I bond direction), the total energy decreases while the dipole moment increases. The C-I and C-H bond lengths increase gradually. The energy gap first increases and then decreases with the external electric field. Further studies show that when the external electric field increases from 0 to 0.04 a.u., the dissociation PES along the C-I bond becomes unbound with the potential energy barrier disappearing. The external electric field of 0.04 a.u. is sufficient to induce the degradation of methyl iodide with the C-I bond breaking. This work provides an important support for the degradation of methyl iodide in the external electric field.

We compute four-dimensional diabatic potential energy surfaces and transition dipole moment surfaces of O2-O2, relevant for the theoretical description of collision-induced absorption in the forbidden X3Σg- → a1Δg and X3Σg- → b1Σg+ bands at 7883 cm-1 and 13 122 cm-1, respectively. We compute potentials at the multi-reference configuration interaction (MRCI) level and dipole surfaces at the MRCI and complete active space self-consistent field (CASSCF) levels of theory. Potentials and dipole surfaces are transformed to a diabatic basis using a recent multiple-property-based diabatization algorithm. We discuss the angular expansion of these surfaces, derive the symmetry constraints on the expansion coefficients, and present working equations for determining the expansion coefficients by numerical integration over the angles. We also present an interpolation scheme with exponential extrapolation to both short and large separations, which is used for representing the O2-O2 distance dependence of the angular expansion coefficients. For the triplet ground state of the complex, the potential energy surface is in reasonable agreement with previous calculations, whereas global excited state potentials are reported here for the first time. The transition dipole moment surfaces are strongly dependent on the level of theory at which they are calculated, as is also shown here by benchmark calculations at high symmetry geometries. Therefore, ab initio calculations of the collision-induced absorption spectra cannot become quantitatively predictive unless more accurate transition dipole surfaces can be computed. This is left as an open question for method development in electronic structure theory. The calculated potential energy and transition dipole moment surfaces are employed in quantum dynamical calculations of collision-induced absorption spectra reported in Paper II [T. Karman et al., J. Chem. Phys. 147, 084307 (2017)].

It is demonstrated that the compounds of late actinides, namely Lawrencium and Nobelium surprisingly exhibit unusual nonactinide properties in that unlike other actinides the chemistry of these species is principally determined by the 7s and 7p orbitals rather than the 5f or 6d shells. Relativistic computations including electron correlation and spin–orbit effects using the complete-active space multiconfiguration self-consistent field followed by second-order and multireference relativistic configuration interaction (RCI) techniques are considered for the Lawrencium and Nobelium dihydrides as well the atoms. The ground and first excited states of Lawrencium and Nobelium arise from the 7s and 7p shells, and thus the potential energy surfaces of these species are unusual in having considerable 7p characteristics. Both molecules form stable bent ground states reminiscent of sp2 hybridization with equilibrium bond angles near 120°. The Lawrencium compounds exhibit unusual characteristics due to avoided crossings of the potential energy surfaces. As a result of spin–orbit coupling, the B22 state of LrH2 undergoes avoided crossing with the A12 state in the spin double group, which reduces the barrier for insertion of Lr into H2. The Nobelium compounds are shown to be considerably less stable compared to Lawrencium compounds due to the relativistic stabilization of the 7s shell of the Nobelium atom. It is shown that the barrier for insertion of Lr into H2 is lowered by relativity (spin–orbit coupling), while No has to surpass a larger barrier due to the relativistic stabilization of the 7s2 shell, which is not very reactive. Lawrencium is the only element in the actinide series with unusually low ionization potential, and NoH2 has an unusually large dipole moment of 5.9 Debye. It is suggested that the Lawrencium and Nobelium compounds should have periodic similarities to the thallium and radium compounds, respectively.

In local effective potential energy theories such as the Hohenberg-Kohn-Sham density functional theory (HKS-DFT) and quantal density functional theory (Q-DFT), electronic systems in their ground or excited states are mapped to model systems of noninteracting fermions with equivalent density. From these models, the equivalent total energy and ionization potential are also obtained. This paper concerns (i) the nonuniqueness of the local effective potential energy function of the model system in the mapping from a nondegenerate ground state, (ii) the nonuniqueness of the local effective potential energy function in the mapping from a nondegenerate excited state, and (iii) in the mapping to a model system in an excited state, the nonuniqueness of the model system wave function. According to nondegenerate ground state HKS-DFT, there exists only one local effective potential energy function, obtained as the functional derivative of the unique ground state energy functional, that can generate the ground state density. Since the theorems of ground state HKS-DFT cannot be generalized to nondegenerate excited states, there could exist different local potential energy functions that generate the excited state density. The constrained-search version of HKS-DFT selects one of these functions as the functional derivative of a bidensity energy functional. In this paper, the authors show via Q-DFT that there exist an infinite number of local potential energy functions that can generate both the nondegenerate ground and excited state densities of an interacting system. This is accomplished by constructing model systems in configurations different from those of the interacting system. Further, they prove that the difference between the various potential energy functions lies solely in their correlation-kinetic contributions. The component of these functions due to the Pauli exclusion principle and Coulomb repulsion remains the same. The existence of the different potential energy functions as viewed from the perspective of Q-DFT reaffirms that there can be no equivalent to the ground state HKS-DFT theorems for excited states. Additionally, the lack of such theorems for excited states is attributable to correlation-kinetic effects. Finally, they show that in the mapping to a model system in an excited state, there is a nonuniqueness of the model system wave function. Different wave functions lead to the same density, each thereby satisfying the sole requirement of reproducing the interacting system density. Examples of the nonuniqueness of the potential energy functions for the mapping from both ground and excited states and the nonuniqueness of the wave function are provided for the exactly solvable Hooke's atom. The work of others is also discussed.

The full configuration interaction quantum Monte Carlo (FCIQMC) is a state-of-the-art stochastic electronic structure method, providing a methodology to compute FCI-level state energies of molecular systems within a quantum chemical basis. However, especially to probe dynamics at the FCIQMC level, it is necessary to devise more efficient schemes to produce nuclear forces and potential energy surfaces (PES) from FCIQMC. In this work, we derive the general formula for nuclear forces from FCIQMC, and clarify different contributions of the total force. This method to obtain FCIQMC forces eliminates previous restrictions and can be used with frozen core approximation and free selection of orbitals, making it promising for more efficient nuclear forces calculations. After some numerical checks of this procedure on the binding curve of N2 molecule, we use the FCIQMC energy and force to obtain the full-dimensional ground state PES of the water molecule via Gaussian processes regression. The new water FCIQMC PES can be used as the basis for H2O ground state nuclear dynamics, structure optimization, and rotation-vibrational spectrum calculation.