Extended Born-Oppenheimer equation for a three-state system

We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrices are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity analytically around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calculations are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.     

Ab initio calculations on the excited states of Na3 cluster to explore beyond Born-Oppenheimer theories: adiabatic to diabatic potential energy surfaces and nuclear dynamics

We perform ab initio calculation using quantum chemistry package (MOLPRO) on the excited states of Na(3) cluster and present the adiabatic PESs for the electronic states 2(2)E' and 1(2)A(1)', and the non-adiabatic coupling (NAC) terms among those states. Since the ab initio calculated NAC elements for the states 2(2)E' and 1(2)A(1)' demonstrate the numerical validity of so called "Curl Condition," such states closely form a sub-Hilbert space. For this subspace, we employ the NAC terms to solve the "adiabatic-diabatic transformation (ADT)" equations to obtain the functional form of the transformation angles and pave the way to construct the continuous and single valued diabatic potential energy surface matrix by exploiting the existing first principle based theoretical means on beyond Born-Oppenheimer treatment. Nuclear dynamics has been carried out on those diabatic surfaces to reproduce the experimental spectrum for system B of Na(3) cluster and thereby, to explore the numerical validity of the theoretical development on beyond Born-Oppenheimer approach for adiabatic to diabatic transformation.     

A rigorous approach to the formulation of extended Born‐Oppenheimer equation for a three‐state system

If a coupled three‐state electronic manifold forms a sub‐Hilbert space, it is possible to express the non‐adiabatic coupling (NAC) elements in terms of adiabatic–diabatic transformation (ADT) angles. Consequently, we demonstrate: (a) Those explicit forms of the NAC terms satisfy the Curl conditions with non‐zero Divergences; (b) The formulation of extended Born‐Oppenheimer (EBO) equation for any three‐state BO system is possible only when there exists coordinate independent ratio of the gradients for each pair of ADT angles leading to zero Curls at and around the conical intersection(s). With these analytic advancements, we formulate a rigorous EBO equation and explore its validity as well as necessity with respect to the approximate one (Sarkar and Adhikari, J Chem Phys 2006, 124, 074101) by performing numerical calculations on two different models constructed with different chosen forms of the NAC elements. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions

An ab initio electronic wave-packet dynamics coupled with the simultaneous classical dynamics of nuclear motions in a molecule is studied. We first survey the dynamical equations of motion for the individual components. Reflecting the nonadiabatic dynamics that electrons can respond to nuclear motions only with a finite speed, the equations of motion for nuclei include a force arising from the kinematic (nuclear momentum) coupling from electron cloud. To materialize these quantum effects in the actual ab initio calculations, we study practical implementation of relevant electronic matrix elements that are related to the derivatives with respect to the nuclear coordinates. Applications of the present scheme are performed in terms of the configuration state functions (CSF) using the canonical molecular orbitals as basis functions without transformation to particular diabatic basis. In the CSF representation, the nonadiabatic interaction due to the kinematic coupling is anticipated to be rather small, and instead it should be well taken into account through the off-diagonal elements of the electronic Hamiltonian matrix. Therefore it is expected that the nonadiabatic dynamics based on this CSF basis neglecting the kinematic coupling may work. To verify this anticipation and to quantify the actual effects of the kinematic coupling, we compare the dynamics with and without the kinematic-coupling terms using the same CSF set. Applications up to the fifth electronically excited states in a nonadiabatic collision between H(2) and B(+) shows that the overall behaviors of these two calculations are surprisingly similar to each other in an average sense except for a fast fluctuation reflecting the electronic time scale. However, at the same time, qualitative differences in the collision events are sometimes observed. Therefore it turns out after all that the kinematic-coupling terms cannot be neglected in the CSF-basis representation. The present applications also demonstrate that the nonadiabatic electronic wave-packet dynamics within ab initio quantum chemical calculation is feasible.     

Two-state versus three-state quantization: an ab initio study of the three lower states of the {N,H2/A'} system

In this article we present the first ab initio study of the conical intersections (cis) and their electronic nonadiabatic coupling terms (NACTs) for the {N,H2} system. Efforts were made to reveal the location of cis between the two lower, 1 2A' and 2 2A' states--to be designated as (1,2) cis--and the cis between the two upper, 2 2A' and 3 2A' states--to be designated as the (2,3) cis--of this system. We found that these cis are located along the collinear {NHH) arrangement. The study is carried out by analyzing two-state magnitudes such as the (1,2) and (2,3) adiabatic-to-diabatic transformation angles (known also as the mixing angles) and the corresponding topological phases (known also as the Berry phases or the Longuet-Higgins phases). In addition, a detailed three-state study is carried out. Here the emphasis is on driving the diagonal elements of the topological D matrix and analyzing situations for which the corresponding nonadiabatic coupling matrix is quantized. The reliability of two-state results is carefully examined by comparing them with corresponding outcomes derived for the three-state study. In addition we also calculated the potential-energy surfaces related to the two lower states and studied to what extent they are affected by the (1,2) ci. The results obtained in this treatment were found to be in full agreement with the NACT's calculations.     

Ab initio surface hopping excited-state molecular dynamics approach on the basis of spin–orbit coupled states: An application to the A-band photodissociation of CH3I

Ab initio molecular dynamics approach has been extended to multi-state dynamics on the basis of the spin–orbit coupled electronic states that are obtained through diagonalization of the spin–orbit coupling matrix with the multi-state second-order multireference perturbation theory energies in diagonal elements and the spin–orbit coupling terms at the state-averaged complete active space self-consistent field level in off-diagonal elements. Nonadiabatic transitions over the spin–orbit coupled states were taken into account explicitly by a surface hopping scheme with utilizing the nonadiabatic coupling terms calculated by numerical differentiation of the spin–orbit coupled wavefunctions and analytical nonadiabatic coupling terms. The present method was applied to the A-band photodissociation of methyl iodide, CH₃I + hv → CH₃ + I (²P_3/2)/I* (²P_1/2), for which a pioneering theoretical work was reported by Amatatsu, Yabushita, and Morokuma. The present results reproduced well the experimental branching ratio and energy distributions in the dissociative products. © 2018 Wiley Periodicals, Inc.     

Derivation of the electronic nonadiabatic coupling field in molecular systems: an algebraic-vectorial approach

In this Communication it is suggested that various elements of the nonadiabatic coupling matrix, tau(jk)(s) are created by the singular nonadiabatic coupling terms of the system. Moreover, given the spatial distribution of these coupling terms in the close vicinity of their singularity points yields, according to this approach, the integrated intensity of the field at every point in the region of interest. To support these statements we consider the conical intersections of the three lower states of the H+H(2) system: From an ab initio treatment we obtain the nonadiabatic coupling terms around each conical intersection separately (at its close vicinity) and having those, create the field at every desired point employing vector-algebra. This approach is also used to calculate the intensity of the Curl of those matrix elements that lack their own sources [tau(13)(s) in the present case]. The final results are compared with relevant ab initio calculations.     

Analytical determination of non-born—oppenheimer matrix elements between different molecular electronic states

Matrix elements for the nonadiabatic correction between different adiabatic Born—Oppenheimer (ABO) states are discussed in terms of a generating function (nonadiabatic coupling function, NAF) which was introduced earlier. Recursion relations are derived for the case where different vibronic states are coupled via a single non-totally symmetric mode. The formalism is applied to the estimation of non-Born—Oppenheimer coupling between vibronic states of pyrazine.     

Ab initio adiabatic and quasidiabatic potential energy surfaces of lowest four electronic states of the H+ + O2 system

Ab initio global adiabatic and quasidiabatic potential energy surfaces of lowest four electronic (1-4 (3)A(")) states of the H(+)+O(2) system have been computed in the Jacobi coordinates (R,r,γ) using Dunning's cc-pVTZ basis set at the internally contracted multireference (single and double) configuration interaction level of accuracy, which are relevant to the dynamics studies of inelastic vibrational and charge transfer processes observed in the scattering experiments. The computed equilibrium geometry parameters of the bound [HO(2)](+) ion in the ground electronic state and other parameters for the transition state for the isomerization process, HOO(+)?OOH(+) are in good quantitative agreement with those available from the high level ab initio calculations, thus lending credence to the accuracy of the potential energy surfaces. The nonadiabatic couplings between the electronic states have been analyzed in both the adiabatic and quasidiabatic frameworks by computing the nonadiabatic coupling matrix elements and the coupling potentials, respectively. It is inferred that the dynamics of energy transfer processes in the scattering experiments carried out in the range of 9.5-23 eV would involve all the four electronic states.     

Theoretical spectroscopy of the calcium dimer in the A 1Sigma(u)+, c3Pi(u), and a3Sigma(u)+ manifolds: an ab initio nonadiabatic treatment

Nonadiabatic theory of molecular spectra of diatomic molecules is presented. It is shown that in the fully nonadiabatic framework, the rovibrational wave functions describing the nuclear motions in diatomic molecules can be obtained from a system of coupled differential equations. The rovibrational wave functions corresponding to various electronic states are coupled through the relativistic spin-orbit coupling interaction and through different radial and angular coupling terms, while the transition intensities can be written in terms of the ground state rovibrational wave function and bound rovibrational wave functions of all excited electronic states that are electric dipole connected with the ground state. This theory was applied in the nearly exact nonadiabatic calculations of energy levels, line positions, and intensities of the calcium dimer in the A (1)Sigma(u) (+)(1 (1)S+1 (1)D), c (3)Pi(u)(1 (3)P+1 (1)S), and a (3)Sigma(u) (+)(1 (3)P+1 (1)S) manifolds of states. The excited state potentials were computed using a combination of the linear response theory within the coupled-cluster singles and doubles framework for the core-core and core-valence electronic correlations and of the full configuration interaction for the valence-valence correlation, and corrected for the one-electron relativistic terms resulting from the first-order many-electron Breit theory. The electric transition dipole moment governing the A (1)Sigma(u) (+)<--X (1)Sigma(g) (+) transitions was obtained as the first residue of the frequency-dependent polarization propagator computed with the coupled-cluster method restricted to single and double excitations, while the spin-orbit and nonadiabatic coupling matrix elements were computed with the multireference configuration interaction wave functions restricted to single and double excitations. Our theoretical results explain semiquantitatively all the features of the observed Ca(2) spectrum in the A (1)Sigma(u) (+)(1 (1)S+1 (1)D), c (3)Pi(u)(1 (3)P+1 (1)S), and a (3)Sigma(u) (+)(1 (3)P+1 (1)S) manifolds of states.     

Full six-dimensional nonadiabatic quantum dynamics calculation for the energy pooling reaction O(2)(a (1)Delta)+O(2)(a (1)Delta)-->O(2)(b (1)Sigma)+O(2)(X (3)Sigma)

Six new potential energy surfaces of four singlet states and two triplet states for the title oxygen molecule reaction along with the spin-orbit coupling among them have been constructed from the complete active space second-order perturbation theory with a 6-311+G(d) basis. Accurate integral cross sections are calculated with a full six-dimensional nonadiabatic time-dependent quantum wave packet method. The thermal rate constant based on the integral cross sections agrees well with the result of the experimental measurements, and the intersystem crossing effects are also discussed in this electronic energy-transfer process.     

Existence of strictly diabatic basis sets for the two‐state problem

Some new properties of the nonadiabatic coupling elements are derived, in particular the orthogonality and gauge invariance of their longitudinal and transverse components. A method for constructing a strictly diabatic basis set that makes both the transverse and longitudinal components of the nonadiabatic coupling elements of the two‐state problem vanish identically and is based on introducing overlap between the electronic states in the vicinity of the crossing seam is proposed. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Nonadiabatic scattering of NO off Au3 clusters: A simple and robust diabatic state manifold generation method for multiconfigurational wavefunctions

The presence of nonadiabatic effects during the interaction of small molecules with metals has been observed experimentally for the last decades. Specially remarkable are the effects found for NO/Au, where experiments have suggested the presence of very strong vibronic coupling during the molecular scattering. However, the accurate inclusion of the nonadiabatic effects in periodic boundary conditions (PBC) theoretical methods remain an unapproachable challenge. Here, aiming to give some theoretical insight to the strong vibronic coupling, we have adopted a pragmatic point of view, taking use of an auxiliary simplified system, NO/Au₃. We show the importance of nonadiabatic coupling, during the scattering of NO from a Au₃ cluster, using a diabatic representation of 12 electronic states of the system, including a few charge-transfer states. Our diabatic representation is obtained by rotating the orbital and configuration interaction (CI) vectors of a restricted active space (RAS) wavefunction. We present a strategy for extracting the best effective manifold of states relevant to the system, below some prescribed energy, directly from the RAS CI vectors. This scheme is able to disentangle a large dense manifold of adiabatic states with strong coupling and crossings. This approach is also shown to work for multireference configuration interaction (MRCI). By performing quantum propagations, we observed an increase in vibrational redistribution with increasing initial vibrational or translational energies. We suggest that these nonadiabatic effects should also be present at smaller energies in larger clusters. © 2018 Wiley Periodicals, Inc.     

Time‐dependent molecular fields created by the interaction of an external electromagnetic field with a molecular system

When an external, time‐dependent field interacts with a molecular system various phenomena may take place. However, concentrating on a region close enough to a point of conical intersection, we find that this external field builds up a field similar to an electromagnetic field formed on the one hand by Field‐Dressed nonadiabatic coupling terms which are reminiscent of the Maxwell–Lorentz Vector potentials, and on the other hand via a scalar potential formed by the dipole‐interaction with an external field. In this article, we show that this new field, to be termed Molecular Field, is characterized by several spatial and space‐time Field‐Dressed Curl equations and one, single, space‐time Field‐Dressed Divergence equation. These equations are then shown to yield, just as in the general theory of electromagnetism, the corresponding Field‐Dressed Wave Equations. This achievement could be materialized employing the (1,2) antisymmetric matrix elements of any of the 2×2 dimensional Field‐Dressed nonadiabatic coupling matrices. © 2014 Wiley Periodicals, Inc.     

Proton-coupled electron transfer versus hydrogen atom transfer: generation of charge-localized diabatic states

The distinction between proton-coupled electron transfer (PCET) and hydrogen atom transfer (HAT) mechanisms is important for the characterization of many chemical and biological processes. PCET and HAT mechanisms can be differentiated in terms of electronically nonadiabatic and adiabatic proton transfer, respectively. In this paper, quantitative diagnostics to evaluate the degree of electron-proton nonadiabaticity are presented. Moreover, the connection between the degree of electron-proton nonadiabaticity and the physical characteristics distinguishing PCET from HAT, namely, the extent of electronic charge redistribution, is clarified. In addition, a rigorous diabatization scheme for transforming the adiabatic electronic states into charge-localized diabatic states for PCET reactions is presented. These diabatic states are constructed to ensure that the first-order nonadiabatic couplings with respect to the one-dimensional transferring hydrogen coordinate vanish exactly. Application of these approaches to the phenoxyl-phenol and benzyl-toluene systems characterizes the former as PCET and the latter as HAT. The diabatic states generated for the phenoxyl-phenol system possess physically meaningful, localized electronic charge distributions that are relatively invariant along the hydrogen coordinate. These diabatic electronic states can be combined with the associated proton vibrational states to generate the reactant and product electron-proton vibronic states that form the basis of nonadiabatic PCET theories. Furthermore, these vibronic states and the corresponding vibronic couplings may be used to calculate rate constants and kinetic isotope effects of PCET reactions.     

Non-Born-Oppenheimer trajectories with self-consistent decay of mixing

A semiclassical trajectory method, called the self-consistent decay of mixing (SCDM) method, is presented for the treatment of electronically nonadiabatic dynamics. The SCDM method is a modification of the semiclassical Ehrenfest (SE) method (also called the semiclassical time-dependent self-consistent-field method) that solves the problem of unphysical mixed final states by including decay-of-mixing terms in the equations for the evolution of the electronic state populations. These terms generate a force, called the decoherent force (or dephasing force), that drives the electronic component of each trajectory toward a pure state. Results for several mixed quantum-classical methods, in particular the SCDM, SE, and natural-decay-of-mixing methods and several trajectory surface hopping methods, are compared to the results of accurate quantum mechanical calculations for 12 cases involving five different fully dimensional triatomic model systems. The SCDM method is found to be the most accurate of the methods tested. The method should be useful for the simulation of photochemical reactions.     

Quantized time correlation function approach to nonadiabatic decay rates in condensed phase: application to solvated electrons in water and methanol

A new, alternative form of the golden rule formula defining the nonadiabatic transition rate between two quantum states in condensed phase is presented. The formula involves the quantum time correlation function of the energy gap, of the nonadiabatic coupling, and their cross terms. Those quantities can be inferred from their classical counterparts, determined via molecular dynamics simulations. The formalism is applied to the problem of the nonadiabatic p-->s relaxation of an equilibrated p-electron in water and methanol. We find that, in both solvents, the relaxation is induced by the coupling to the vibrational modes and the quantum effects modify the rate by a factor of 2-10 depending on the quantization procedure applied. The resulting p-state lifetime for a hypothetical equilibrium excited state appears extremely short, in the sub-100 fs regime. Although this result is in contrast with all previous theoretical predictions, we also illustrate that the lifetimes computed here are very sensitive to the simulated electronic quantum gap and to the strongly correlated nonadiabatic coupling.     

Assigning signs to the electronic nonadiabatic coupling terms: the H2,O system as a case study

This paper is devoted to a specific difficulty related to the electronic nonadiabatic coupling terms (NACT), namely, how to determine correctly their signs. It is well known that correct NACTs, including their signs, are crucial for any numerical treatment of the nuclear Schrodinger equation [see, i.e., A. Kuppermaan and R. Abrol, Adv. Chem. Phys. 124, 283 (2003)]. In most cases the derivation of the correct sign of the nonadiabatic coupling matrix (NACM) is done employing various continuity procedures. However, there are cases where these procedures do not suffice and for these cases we suggest to apply an additional procedure based on a mathematical lemma which asserts that the exponentiated line integral which yields the D matrix is invariant with respect to the initial point of the integration [M. Baer, J. Phys. Chem. A 104, 3181 (2000)]. In the numerical study we apply this lemma to determine the signs of the 3x3 NACM elements for the three excited states of the {H(2),O} system (some of these NACTs are presented here for the first time). It turns out that the ab initio treatment yields results from which one can form eight different 3x3 NACMs. However the application of this lemma (which does not require any significant additional numerical effort) reduces this number to two. The final selection is done by an enhanced numerical study which requires more accurate calculations.     

High resolution vacuum ultraviolet emission spectrum of D(2) from 78 to 103 nm: The D (1)Pi(u)-->X (1)Sigma(g) (+) and D(') (1)Pi(u) (-)-->X (1)Sigma(g) (+) band systems

The emission spectrum of the D(2) molecule has been studied at high resolution in the vacuum ultraviolet region 78.5-102.7 nm. A detailed analysis of the two D (1)Pi(u)-->X (1)Sigma(g) (+) and D(') (1)Pi(u) (-)-->X (1)Sigma(g) (+) electronic band systems is reported. New and improved values of the level energies of the two upper states have been derived with the help of the program IDEN [V. I. Azarov, Phys. Scr. 44, 528 (1991); 48, 656 (1993)], originally developed for atomic spectral analysis. A detailed comparison is made between the observed energy levels and solutions of coupled equations using the newest ab initio potentials by Wolniewicz and co-workers [J. Chem. Phys. 103, 1792 (1995); 99, 1851 (1993); J. Mol. Spectros. 212, 208 (2002); 220, 45 (2003)] taking into account the nonadiabatic coupling terms for the D (1)Pi(u) state with the lowest electronic states B (1)Sigma(u) (+), C (1)Pi(u), and B(') (1)Sigma(u) (+). A satisfactory agreement has been found for most of the level energies belonging to the D and D(') states. The remaining differences between observation and theory are probably due to nonadiabatic couplings with other higher electronic states which were neglected in the calculations.     

Curl equations as substratum for the derivation of the electronic nonadiabatic coupling terms

The idea to derive the nonadiabatic coupling terms by solving the Curl equations (Avery, J.; Baer, M.; Billing, G. D. Mol Phys 2002, 100, 1011) is extended to a three‐state system where the first and second states form one conical intersection, i.e., τ₁₂ and the second and the third states form another conical intersection, i.e., τ₂₃. Whereas the two‐state Curl equations form a set of linear differential equations, the extension to a three‐state system not only increases the number of equations but also leads to nonlinear terms. In the present study, we developed a perturbative scheme, which guarantees convergence if the overlap between the two interacting conical intersections is not too strong. Among other things, we also revealed that the nonadiabatic coupling term between the first and third states, i.e., τ₁₃ (such interactions do not originate from conical intersection) is formed due to the interaction between τ₁₂ and τ₂₃. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002