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We construct the effective Hartree potential for H2 on Cu(111) as introduced in our earlier work [Dutta et al., J. Chem. Phys. 154, 104103 (2021), and Dutta et al., J. Chem. Phys. 157, 194112 (2022)] starting from the same gas–metal interaction potential obtained for 0 K. Unlike in that work, we now explicitly account for surface expansion at 925 K and investigate different models to describe the surface vibrational modes: (i) a cluster model yielding harmonic normal modes at 0 K and (ii) slab models resulting in phonons at 0 and 925 K according to the quasi-harmonic approximation—all consistently calculated at the density functional theory level with the same exchange–correlation potential. While performing dynamical calculations for the H2(v = 0, j = 0)–Cu(111) system employing Hartree potential constructed with 925 K phonons and surface temperature, (i) the calculated chemisorption probabilities are the highest compared to the other approaches over the energy domain and (ii) the threshold for the reaction probability is the lowest, in close agreement with the experiment. Although the survival probabilities (v′ = 0) depict the expected trend (lower in magnitude), the excitation probabilities (v′ = 1) display a higher magnitude since the 925 K phonons and surface temperature are more effective for the excitation process compared to the phonons/normal modes obtained from the other approaches investigated to describe the surface. [ABSTRACT FROM AUTHOR]

While carrying out Beyond Born–Oppenheimer theory based diabatization, the solutions of adiabatic-to-diabatic transformation equations depend on the paths of integration over two-dimensional cross-sections of multi-dimensional space of nuclear degrees of freedom. It is shown that such path-dependent solutions leading to diabatic potential energy surface matrices computed along any two different paths are related through an orthogonal matrix, and thereby, those surface matrices should provide unique observables. While exploring the numerical validity of the theoretical framework, we construct diabatic Hamiltonians for the five low-lying electronic states ( X ̃ 2 E 1 g , B ̃ 2 E 2 g , and C ̃ 2 A 2 u ) of benzene radical cation ( C 6 H 6 + ) along three different approaches of contour integration over two dimensional nuclear planes constituted by seven non-adiabatically active normal modes. Three different diabatic surface matrices are further employed to generate the photoelectron spectra of the benzene molecule (C6H6). It is interesting to note that the spectral peak positions and intensity patterns for all three cases are almost close to each other and also exhibit very good agreement with the experimental results. [ABSTRACT FROM AUTHOR]

We have carried out fully close-coupled three dimensional quantum mechanical wave packet dynamical calculations for the reaction He + H 2 + → HeH + + H on the ground electronic adiabatic potential energy surface and on the lowest two electronic states of newly constructed ab initio calculated diabatic potential energy surfaces for the system [Naskar et al., J. Phys. Chem. A 127, 3832 (2023)]. With the reactant diatom ( H 2 + ) in its roto-vibrational ground state (v = 0, j = 0), the calculations have been carried out in hyperspherical coordinates to obtain the reaction attributes. Convergence profiles of the reaction probability with respect to the total angular momentum quantum number at different collision energies are presented for the title reaction. State-to-state as well as initial state selected integral reaction cross sections are calculated from the fully converged reaction probabilities over a range of collision energies. The integral cross section values computed using the two-state diabatic potential energy surfaces are significantly lower than those obtained using the ground electronic state adiabatic potential energy surface and are in much better agreement with the available experimental results than the latter for total energy greater than 1.1 eV. Therefore, it becomes clear that it is important to include the nonadiabatic coupling terms for a quantitative prediction of the dynamical observables. [ABSTRACT FROM AUTHOR]

The effect of surface mode vibrations on the reactive scattering of D2, initialized in the ground rovibrational state (v = 0, j = 0), from a Cu(111) surface is investigated for different surface temperature situations. We adopt a time and temperature dependent effective Hamiltonian [Dutta et al., J. Chem. Phys. 154, 104103 (2021)] constructed by combining the linearly coupled many oscillator model [Sahoo et al., J. Chem. Phys. 136, 084306 (2012)] and the static corrugation model [M. Wijzenbroek and M. F. Somers, J. Chem. Phys. 137, 054703 (2012)] potential within the mean-field approach. Such an effective Hamiltonian is employed for six-dimensional quantum dynamical calculations to obtain temperature dependent reaction and state-to-state scattering probability profiles as a function of incidence energy of colliding D2 molecules. As reported in the experimental studies, the movements of surface atoms modify the dissociative scattering dynamics at higher surface temperature by exhibiting vibrational quantum and surface atoms' recoil effects in the low and high collision energy domains, respectively. Finally, we compare our present theoretical results with the experimental and other theoretical outcomes, as well as discuss the novelty of our findings. [ABSTRACT FROM AUTHOR]

The effect of surface atom vibrations on H2 scattering from a Cu(111) surface at different temperatures is being investigated for hydrogen molecules in their rovibrational ground state (v = 0, j = 0). We assume weakly correlated interactions between molecular degrees of freedom and surface modes through a Hartree product type wavefunction. While constructing the six-dimensional effective Hamiltonian, we employ (a) a chemically accurate potential energy surface according to the static corrugation model [M. Wijzenbroek and M. F. Somers, J. Chem. Phys. 137, 054703 (2012)]; (b) normal mode frequencies and displacement vectors calculated with different surface atom interaction potentials within a cluster approximation; and (c) initial state distributions for the vibrational modes according to Bose–Einstein probability factors. We carry out 6D quantum dynamics with the so-constructed effective Hamiltonian and analyze sticking and state-to-state scattering probabilities. The surface atom vibrations affect the chemisorption dynamics. The results show physically meaningful trends for both reaction and scattering probabilities compared to experimental and other theoretical results. [ABSTRACT FROM AUTHOR]

We employ theoretically "exact" and numerically "accurate" Beyond Born–Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation ( C 6 H 6 + ) for the first time and explore the workability of the time-dependent discrete variable representation (TDDVR) method for carrying out dynamical calculations to evaluate the photoelectron (PE) spectra of its neutral analog. Ab initio adiabatic PESs and nonadiabatic coupling terms are computed over a series of pairwise normal modes, which exhibit rich nonadiabatic interactions starting from Jahn–Teller interactions and accidental conical intersections/seams to pseudo Jahn–Teller couplings. Once the electronic structure calculation is completed on the low-lying five doublet electronic states ( X ̃ 2 E 1 g , B ̃ 2 E 2 g , and C ̃ 2 A 2 u ) of the cationic species, diabatization is carried out employing the adiabatic-to-diabatic transformation (ADT) equations for the five-state sub-Hilbert space to compute highly accurate ADT angles, and thereby, single-valued, smooth, symmetric, and continuous diabatic PESs and couplings are constructed. Subsequently, such surface matrices are used to perform multi-state multi-mode nuclear dynamics for simulating PE spectra of benzene. Our theoretical findings clearly depict that the spectra for X ̃ 2 E 1 g and B ̃ 2 E 2 g − C ̃ 2 A 2 u states obtained from BBO treatment and TDDVR dynamics exhibit reasonably good agreement with the experimental results as well as with the findings of other theoretical approaches. [ABSTRACT FROM AUTHOR]

First principles based beyond Born–Oppenheimer theory has been implemented on the F + H2 system for constructing multistate global diabatic Potential Energy Surfaces (PESs) through the incorporation of Nonadiabatic Coupling Terms (NACTs) explicitly. The spin–orbit (SO) coupling effect on the collision process of the F + H2 reaction has been included as a perturbation to the non-relativistic electronic Hamiltonian. Adiabatic PESs and NACTs for the lowest three electronic states (12A′, 22A′, and 12A″) are determined in hyperspherical coordinates as functions of hyperangles for a grid of fixed values of the hyperradius. Jahn–Teller (JT) type conical intersections between the two A′ states translate along C2 v and linear geometries in F + H2. In addition, A′ and A″ states undergo Renner–Teller (RT) interaction at collinear configurations of this system. Both JT and RT couplings are validated by integrating NACTs along properly chosen contours. Subsequently, we have solved adiabatic-to-diabatic transformation (ADT) equations to evaluate the ADT angles for constructing the diabatic potential matrix of F + H2, including the SO coupling terms. The newly calculated diabatic PESs are found to be smooth, single-valued, continuous, and symmetric and can be invoked for performing accurate scattering calculations on the F + H2 system. [ABSTRACT FROM AUTHOR]

A model Hamiltonian for centrifugally stabilized electronic-vibrational motion of a cubic perturbed upper "Mexican hat" potential surface for a Mn3+ ion in an octahedral symmetry is formulated, and its eigenspectrum is explored. Theoretically calculated eigenvalues for cubic perturbed ground and excited electronic states are employed to interpret the origin of higher energy narrow side bands (satellite transitions) appearing in the dielectric function spectra of the LaMnO3 complex, which exhibit anomalous temperature dependence in the vicinity of the Néel temperature, TN ≃ 140 K [N. N. Kovaleva et al., J. Exp. Theor. Phys. 122, 890 (2016)]. [ABSTRACT FROM AUTHOR]

In order to circumvent numerical inaccuracy originating from the singularity of nonadiabatic coupling terms (NACTs), we need to perform kinetically coupled adiabatic to potentially coupled diabatic transformation of the nuclear Schrödinger Equation. Such a transformation is difficult to achieve for higher dimensional sub-Hilbert spaces due to inherent complicacy of adiabatic to diabatic transformation (ADT) equations. Nevertheless, detailed expressions of ADT equations are formulated for six coupled electronic states for the first time and their validity is extensively examined for a well-known radical cation, namely, 1,3,5- C 6 H 3 F 3 + (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states ( X ̃ 2 E ′ ′ , Ã 2 A 2 ′ ′ , B ̃ 2 E ′ , and C ̃ 2 A 2 ′ ), where several types of nonadiabatic interactions, like Jahn-Teller conical intersections (CI), accidental CIs, accidental seams (series of degenerate points), and pseudo Jahn-Teller interactions can be observed over the Franck-Condon region of nuclear configuration space. Those interactions are depicted by exploring degenerate components of C–C asymmetric stretching, C–C symmetric stretching, and C–C–C scissoring motion (Q9x, Q9y, Q10x, Q10y, Q12x, and Q12y) to compute complete active space self-consistent field level adiabatic PESs and NACTs as implemented in the MOLPRO quantum chemistry package. Subsequently, we perform the ADT using our newly devised fifteen (15) ADT equations to locate the position of CIs, verify the quantization of NACTs, and to construct highly accurate diabatic PESs. [ABSTRACT FROM AUTHOR]

The workability of beyond Born-Oppenheimer theory to construct diabatic potential energy surfaces (PESs) of a charge transfer atom-diatom collision process has been explored by performing scattering calculations to extract accurate integral cross sections (ICSs) and rate constants for comparison with most recent experimental quantities. We calculate non-adiabatic coupling terms among the lowest three singlet states of H3+ system (1¹A', 2¹A', and 3¹A') using MRCI level of calculation and solve the adiabatic-diabatic transformation equation to formulate the diabatic Hamiltonian matrix of the same process [S. Mukherjee et al., J. Chem. Phys. 141, 204306 (2014)] for the entire region of nuclear configuration space. The nonadiabatic effects in the D+ +H2 reaction has been studied by implementing the coupled 3D time-dependent wave packet formalism in hyperspherical coordinates [S. Adhikari and A. J. C. Varandas, Comput. Phys. Commun. 184, 270 (2013)] with zero and non-zero total angular momentum (J) on such newly constructed accurate (ab initio) diabatic PESs of H3+. We have depicted the convergence profiles of reaction probabilities for the reactive non-charge transfer, non-reactive charge transfer, and reactive charge transfer processes for different collisional energies with respect to the helicity (K) and total angular momentum (J) quantum numbers. Finally, total and state-to-state ICSs are calculated as a function of collision energy for the initial rovibrational state (v = 0, j = 0) of the H2 molecule, and consequently, those quantities are compared with previous theoretical and experimental results. [ABSTRACT FROM AUTHOR]

A thorough investigation has been performed for electronic structure, topological effect, and nuclear dynamics of NO2 molecule, where the adiabatic potential energy surfaces (PESs), conical intersections between the ground (X²A1) and the first excited state (A²B2), and the corresponding non-adiabatic coupling terms between those states are recalculated [Chem. Phys. 416, 11 (2013)] to achieve enough accuracy in dynamics. We employ beyond Born-Oppenheimer theory for these two state sub-Hilbert space to carry out adiabatic to diabatic transformation (ADT) to obtain the ADT angles and thereby, to construct single-valued, smooth, and continuous diabatic PESs. The analytic expressions for the adiabatic PESs and ADT angles are provided to represent a two-state three-mode diabatic Hamiltonian of NO2 for performing nuclear dynamics to calculate the photoelectron spectra of its anion. It appears that not only Jahn-Teller type coupling but also Renner-Teller interaction contributes significantly on the overall spectrum. The coupling between the electronic states (X²A1 and A²B2) of NO2 is essentially through the asymmetric stretching mode, where the functional form of such interaction is distinctly symmetric and non-linear. [ABSTRACT FROM AUTHOR]

We propose a strategy of using a stochastic optimization technique, namely, simulated annealing to design optimum laser pulses (both IR and UV) to achieve greater fluxes along the two dissociating channels (O18 + O16O16 and O16 + O16O18) in O16O16O18 molecule. We show that the integrated fluxes obtained along the targeted dissociating channel is larger with the optimized pulse than with the unoptimized one. The flux ratios are also more impressive with the optimized pulse than with the unoptimized one. We also look at the evolution contours of the wavefunctions along the two channels with time after the actions of both the IR and UV pulses and compare the profiles for unoptimized (initial) and optimized fields for better understanding the results that we achieve. We also report the pulse parameters obtained as well as the final shapes they take. [ABSTRACT FROM AUTHOR]

A recently proposed coupled three-dimensional time-dependent wave-packet formalism in hyperspherical coordinates is shown to yield accurate results for the reactive non-charge transfer process in the title system at collision energies as low as 100 K, where the lowest sheet of the accurate double many body expansion form for the singlet H+3 is used. The results are compared with available experimental data as well as time-independent calculations, and the agreement shown to be generally good. [ABSTRACT FROM AUTHOR]

We calculate the adiabatic Potential Energy Surfaces (PESs) and the Non-Adiabatic Coupling Terms (NACTs) for the three lowest singlet states of H+3 in hyperspherical coordinates as functions of hyperangles (θ and ɸ) for a grid of fixed values of hyperradius (1.5 ≤ ρ ≤ 20 bohrs) using the MRCI level of methodology employing ab initio quantum chemistry package (MOLPRO). The NACT between the ground and the first excited state translates along the seams on the θ - ɸ space, i.e., there are six Conical Intersections (CIs) at each θ (60° ≤ θ ≤ 90°) within the domain, 0 ≤ ɸ ≤ 2p. While transforming the adiabatic PESs to the diabatic ones, such surfaces show up six crossings along those seams. Our beyond Born-Oppenheimer approach could incorporate the effect of NACTs accurately and construct single-valued, continuous, smooth, and symmetric diabatic PESs. Since the location of CIs and the spatial amplitudes of NACTs are most prominent around ρ = 10 bohrs, generally only those results are depicted. [ABSTRACT FROM AUTHOR]

We perform four-dimensional (4Dx2D) as well as six-dimensional (6D) quantum dynamics on a parametrically time- and temperature-dependent effective Hamiltonian for D2(v, j)-Cu(111) system, where such effective potential has been derived through a mean-field approach between molecular degrees of freedom and surface modes with Bose-Einstein probability factor for their initial state distribution. We present the convergence of the theoretically calculated sticking probabilities employing 4Dx2D quantum dynamics with increasing number of surface atoms as well as layers for rigid surface and the surface at a particular temperature, where the temperature-dependent sticking probabilities appear exclusively dictated by those surface modes directed along the Z-axis. The sticking and state-to-state transition probabilities obtained from 6D quantum dynamics are shown as a function of initial kinetic energy of the diatom at different surface temperature. Theoretically calculated sticking probabilities display the similar trend with the experimentally measured one. [ABSTRACT FROM AUTHOR]

We follow a suggestion by Lipoff and Herschbach [Mol. Phys. 108, 1133 (2010)] and compare dressed and bare adiabatic potentials to get insight regarding the low-energy dynamics (e.g., cold reaction) taking place in molecular systems. In this particular case, we are interested to study the effect of conical intersections (ci) on the interacting atoms. For this purpose, we consider vibrational dressed adiabatic and vibrational dressed diabatic potentials in the entrance channel of reactive systems. According to our study, the most one should expect, in case of F + H2, is a mild effect of the (1, 2) ci on its reactive/exchange process-an outcome also supported by experiment. This happens although the corresponding dressed and bare potential barriers (and the corresponding van der Waals potential wells) differ significantly from each other. [ABSTRACT FROM AUTHOR]

We perform ab initio calculation using quantum chemistry package (MOLPRO) on the excited states of Na3 cluster and present the adiabatic PESs for the electronic states 22E′ and 12A1′, and the non-adiabatic coupling (NAC) terms among those states. Since the ab initio calculated NAC elements for the states 22E′ and 12A1′ 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 Na3 cluster and thereby, to explore the numerical validity of the theoretical development on beyond Born-Oppenheimer approach for adiabatic to diabatic transformation. [ABSTRACT FROM AUTHOR]

The tetra-atomic C2H2+ cation is known to form Renner–Teller-type intersections along its collinear axis. Not too long ago, we studied the nonadiabatic coupling terms (NACTs) of this molecule [G. J. Halász et al., J. Chem. Phys. 126, 154309 (2007)] and revealed two kinds of intersections. (i) By employing one of the hydrogens as a test particle, we revealed the fact that indeed the corresponding (angular) NACTs produce topological (Berry) phases that are equal to 2π, which is a result anticipated in the case of Renner–Teller intersections. (ii) However, to our big surprise, repeating this study when one of the atoms (in this case a hydrogen) is shifted from the collinear arrangement yields for the corresponding topological phase a value that equals π (and not 2π). In other words, shifting (even slightly) one of the atoms from the collinear arrangement causes the intersection to change its character and become a Jahn–Teller intersection. This somewhat unexpected novel result was later further analyzed and confirmed by other groups [e.g., T. Vertesi and R. Englman, J. Phys. B 41, 025102 (2008)]. The present article is devoted to another tetra-atomic molecule, namely, the H2CN molecule, which just like the C2H2+ ion, is characterized by Renner–Teller intersections along its collinear axis. Indeed, we revealed the existence of Renner–Teller intersections along the collinear axis, but in contrast to the C2H2+ case a shift of one atom from the collinear arrangement did not form Jahn–Teller intersections. What we found instead is that the noncollinear molecule was not affected by the shift and kept its Renner–Teller character. Another issue treated in this article is the extension of (the two-state) Berry (topological) phase to situations with numerous strongly interacting states. So far the relevance of the Berry phase was tested for systems characterized by two isolated interacting states, although it is defined for any number of interacting states [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. We intend to show how to overcome this limitation and get a valid, fully justified definition of a Berry phase for an isolated system of any number of interacting states (as is expected). [ABSTRACT FROM AUTHOR]

In this article we consider a molecular system exposed to an intense short-pulsed external field. It is a continuation of a previous publication [A. K. Paul, S. Adhikari, D. Mukhopadhyay et al., J. Phys. Chem. A 113, 7331 (2009)] in which a theory is presented that treats quantum effects due to nonclassical photon states (known also as Fock states). Since these states became recently a subject of intense experimental efforts we thought that they can be treated properly within the existing quantum formulation of dynamical processes. This was achieved by incorporating them in the Born–Oppenheimer (BO) treatment with time-dependent coefficients. The extension of the BO treatment to include the Fock states results in a formidable enhancement in numerical efforts expressed, in particular, in a significant increase in CPU time. In the present article we discuss an approach that yields an efficient and reliable approximation with only negligible losses in accuracy. The approximation is tested in detail for the dissociation process of H2+ as caused by a laser field. [ABSTRACT FROM AUTHOR]

When a set of three states is coupled with each other but shows negligibly weak interaction with other states of the Hilbert space, these states form a sub-Hilbert space. In case of such subspace [J. Chem. Phys. 124, 074101 (2006)], (a) the adiabatic-diabatic transformation (ADT) condition, ∇vector A+τvector A=0 [Chem. Phys. Lett. 35, 112 (1975)], provides the explicit forms of the nonadiabatic coupling (NAC) elements in terms of electronic basis function angles, namely, the ADT angles, and (b) those NAC terms satisfy the so-called curl conditions [Chem. Phys. Lett. 35, 112 (1975)], which ensure the removal of the NAC elements [could be singular also at specific point(s) or along a seam in the configuration space] during the ADT to bring the diabatic representation of the nuclear Schrödinger equation with a smooth functional form of coupling elements among the electronic states. Since the diabatic to adiabatic representation of the Hamiltonian is related through the same unitary transformation (∇A+τA=0), it could be quite interesting to explore the nature of the nonadiabatic coupling terms starting from a diabatic Hamiltonian and, thereafter, to formulate the extended Born–Oppenheimer (EBO) equation for those adiabatic states transformed from diabatic ones. We consider a three-state diabatic potential matrix constructed for the excited states of Na3 cluster [J. Chem. Phys. 88, 6068 (1988)] at the pseudo-Jahn–Teller model situation, which can reproduce experimentally measured vibrationally resolved absorption lines [Surf. Sci. 156, 770 (1985)] with appropriate choice of coupling parameters, analytically calculate the nonadiabatic coupling elements along with their curls, and numerically evaluate the ADT angles to explore the nature of its nonadiabaticity. While formulating the single surface beyond the BO equation, our theoretical derivation demonstrates that the existence of zero curls of the NAC terms is a necessity. Indeed, when the energy gap between the third state (12 A1′/22 A1′) and the doubly degenerate states (22 E′/32 E′) of the model Hamiltonian for Na3 cluster is considered to be either identically or approximately zero, the curl for each NAC element naturally approaches zero, leading to a theoretically valid EBO equation. We demonstrate the numerical validity of the EBO equation by calculating the nonadiabatic effects on the photoabsorption spectrum starting with the initial wave function located on the ground electronic state and compare with the corresponding diabatic spectrum when the three states are either degenerate at a point or approaching to form three-state degeneracy at the same point. Finally, we calculate the vibrational eigenspectrum of the ground adiabatic state by using (so to say) theoretically and numerically valid EBO equation to compare with those experimentally measured and BO/geometric phase calculated spectra (Tables I-III). [ABSTRACT FROM AUTHOR]

We demonstrate the workability of a parallelized algorithm of the time-dependent discrete variable representation (TDDVR) method to explore the detailed dynamical aspects of vibronic interaction in two three-state model Hamiltonians (X 2E1g, B 2E2g, C 2A2u and B 2E2g, D 2E1u, E 2B2u) of benzene radical cation along with a preliminary investigation on its five electronic states (X 2E1g, B 2E2g, C 2A2u, D 2E1u, and E2B2u). Since those electronic states are interconnected through a series of conical intersections, we have used six and nine vibronically important modes for the three- and five-state Hamiltonians, respectively, in order to perform the quantum dynamics on such system. The population profiles calculated by using our TDDVR approach show reasonably good agreement with the results obtained by exact quantum mechanical (multiconfiguration time-dependent Hartree) method, whereas the corresponding (calculated) photoabsorption spectra originating from various electronic states agree well with the experimental ones. It is important to note that the parallelized algorithm of our TDDVR approach reduces the computation cost by more than an order of magnitude compared to its serial analog. The TDDVR approach appears to be a good compromise between accuracy and speed for such large molecular system, where quantum mechanical description is needed in a restricted region. [ABSTRACT FROM AUTHOR]

This second article in the two back-to-back articles presents a numerical application to support and strengthen two theoretical findings extensively discussed in the previous article (article I). In I, we found that introducing the space-time contours enables to distinguish between N, the number of nuclear Schrödinger equations to be solved, and L, the number of field-free states that become populated by the external field (in the ordinary, perturbative approaches this distinction is not apparent). In the numerical study we show, employing the electronic transition probability matrix P(s,t) [which closely is related to the transformation matrix ω(s,t)—see Eqs. (21) and (25) in I], that the N=L case is rare and in most cases we have N≤L. Since the perturbative approach can be shown to follow when N=L (see Sec. III C in I) the numerical study implies that in most cases the perturbative approach is not reliable. The second issue that is studied is related to the diabatization process. It is shown, numerically, that the N≥L case, in general, does not lead to field-dressed diabatic potentials which are single valued. However, if N is chosen to be identical to the number of field-free states that yield field-free single-valued diabatic potentials in a given spatial region then the corresponding N field-dressed states also yield single-valued (field-dressed) diabatic potentials. This result is independent of L. The numerical study is carried out for an eigenvalue problem based on the Mathieu equation. [ABSTRACT FROM AUTHOR]

A molecular system exposed to an intense external field is considered. The strength of the field is measured by the number L of electronic states that become populated during this process. In the present article the authors discuss a rigorous way, based on the recently introduced space-time contours [R. Baer, et al., J. Chem. Phys. 119, 6998 (2003)], to form N coupled Schrödinger equations where N≤L, which maintains the effects due to the remaining (L-N) populated states. It is shown that whereas the size of L is unlimited, the main requirement concerning N is that the original group of N field-free states forms a Hilbert subspace in the spatial region of interest. From previous studies it is known that a group of states forms a Hilbert subspace if and only if the corresponding topological D matrix is diagonal [M. Baer, et al., Farad, Discuss 127, 337 (2004)] [ABSTRACT FROM AUTHOR]

In this article the authors relate the possibility that an intense electric field affects topological features of a molecular system. For this purpose they studied a model based on the Mathieu equation. They found that such a field may affect the spatial distribution of the nonadiabatic coupling terms but not the position of the intersections. In other words an intense electric field does not create or annihilate conical intersections. It is shown that this conclusion is valid as long as the field is an analytic function of the coordinates in the region of interest. These findings can be extended to magnetic fields (or electromagnetic fields) as long as they are analytic functions in the region of interest. [ABSTRACT FROM AUTHOR]

We investigate the molecular dynamics of pyrazine after excitation to the S2 electronic state by using the time-dependent discrete variable representation (TDDVR) method. The investigation has been carried out with a realistic 24-mode model Hamiltonian consisting of all the vibrational degrees of freedom of pyrazine molecule. First, we perform the simulation on a basic four-mode model, and then by including additional eight important modes and finally, by introducing 20 bath modes on the basic model. This sequential inclusion of bath modes demonstrates the effect of weak modes on the subsystem, where the calculations of energy and population transfer from basic model to the bath quantify the same effect. The spectral profile obtained by using TDDVR approach shows reasonably good agreement with the results calculated by quantum mechanical approach. It appears that the TDDVR approach for those large systems where quantum mechanical description is needed in a restricted region is a good compromise between accuracy and speed. [ABSTRACT FROM AUTHOR]

We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrödinger 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. [ABSTRACT FROM AUTHOR]

We have used the time-dependent discrete variable representation (TDDVR) method to simulate the photoabsorption spectrum of pyrazine. The time-dependent molecular dynamics of pyrazine after excitation to the S2 electronic state is considered as a benchmark to investigate the S2 absorption spectrum. We have carried out the dynamics on a basic four-mode model of pyrazine with the inclusion of five major modes as well as the rest of the vibrational modes as bath modes. Investigations reveal the effect of bath modes such as energy and population transfer from the subsystem to the bath. Calculated results demonstrate excellent agreement with traditional quantum-mechanical findings during the entire propagation and converge to the exact quantum results when enough gridpoints are used. It appears that TDDVR, as a numerical quantum dynamics methodology, is a good compromise between accuracy and speed. [ABSTRACT FROM AUTHOR]

We demonstrate the workability of a TDDVR based [J. Chem. Phys. 118, 5302 (2003)], novel quantum-classical approach, for simulating scattering processes on a quasi-Jahn–Teller model [J. Chem. Phys. 105, 9141 (1996)] surface. The formulation introduces a set of DVR grid points defined by the Hermite part of the basis set in each dimension and allows the movement of grid points around the central trajectory. With enough trajectories (grid points), the method converges to the exact quantum formulation whereas with only one grid point, we recover the conventional molecular dynamics approach. The time-dependent Schrödinger equation and classical equations of motion are solved self-consistently and electronic transitions are allowed anywhere in the configuration space among any number of coupled states. Quantum-classical calculations are performed on diabatic surfaces (two and three) to reveal the effects of symmetry on inelastic and reactive state-to-state transition probabilities, along with calculations on an adiabatic surface with ordinary Born–Oppenheimer approximation. Excellent agreement between TDDVR and DVR results is obtained in both the representations. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We have investigated the dynamics of a reaction coordinate with or without coupling to a heat bath of harmonic oscillators using a novel time-dependent discrete variable representation (TDDVR) method. The proposed method is semiclassical in nature, theoretically rigorous, and rather straightforward to implement. Excellent agreement of the computed tunneling probabilities and time-averaged tunneling rates with the corresponding exact results demonstrates the efficacy of the proposed approach. Most of the semiclassical calculations reported here have been performed by using classical force, whereas, in a few cases, the quantum force (QF) has been taken into account. It appears that among the TDDVR formulations, it was the first time we have derived a rigorous form of QF from the first principle. [ABSTRACT FROM AUTHOR]

Studies the extended Jahn-Teller model using a time-dependent wave packet approach to see symmetry effects on scattering processes. Symmetry of nuclear wave function due to conical intersections between Born-Oppenheimer potential energy surfaces; Introduction of a vector potential in nuclear Hamiltonian; Incorporation of a phase factor in nuclear wave function.

Demonstrates the Hermite correction method for nonadiabatic transitions. Use of molecular dynamics simulations on a system where electronic transitions are allowed anywhere in configuration space among any number of coupled states; Use of a classical path theory based on the Hermite correction to the Gaussian wave packet expansion; Initialization, propagation, and projection of the wave function.

Focuses on the hyperspherical formulation of the vector potential arising due to the presence of a conical intersection in the adiabatic potential energy hypersurface of an A + B[sub 2] type reactive system. Difference in the absolute peak position of rotational distributions obtained by quasiclassical trajectory calculations.

The selective control of O-H/O-D bond dissociation in reduced dimensionality model of HOD molecule has been explored through IR+UV femtosecond pulses. The IR pulse has been optimized using simulated annealing stochastic approach to maximize population of a desired low quanta vibrational state. Since those vibrational wavefunctions of the ground electronic states are preferentially localized either along the O-H or O-D mode, the femtosecond UV pulse is used only to transfer vibrationally excited molecule to the repulsive upper surface to cleave specific bond, O-H or O-D. While transferring from the ground electronic state to the repulsive one, the optimization of the UV pulse is not necessarily required except specific case. The results so obtained are analyzed with respect to time integrated flux along with contours of time evolution of probability density on excited potential energy surface. After preferential excitation from |0, 0> (|m, n> stands for the state having m and n quanta of excitations in O-H and O-D mode, respectively) vibrational level of the ground electronic state to its specific low quanta vibrational state (|1, 0> or |0, 1> or |2, 0> or |0, 2>) by using optimized IR pulse, the dissociation of O-D or O-H bond through the excited potential energy surface by UV laser pulse appears quite high namely, 88% (O-H ; |1, 0>) or 58% (O-D ; |0, 1>) or 85% (O-H ; |2, 0>) or 59% (O-D ; |0, 2>). Such selectivity of the bond breaking by UV pulse (if required, optimized) together with optimized IR one is encouraging compared to the normal pulses. [ABSTRACT FROM AUTHOR]

Molecular electronic ground-state theories, whether ab initio, or semiempirical are most often formulated as a variational principle, where the electronic ground-state energy, considered a linear or nonlinear functional of a reduced density matrix, obtains a constrained minimum. In this communication, we present a Lagrangian analysis of the self-consistent-field electronic structure problem, which does not resort to the concept of orthogonal molecular orbitals. We also develop a method of constrained minimization efficiently applicable to nonlinear energy functional minimization, as well as to linear models such as tight-binding. The method is able to treat large molecules with an effort that scales linearly with the system size. It has built-in robustness and leads directly to the desired minimal solution. Performance is demonstrated on linear alkane and polyene chains. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We have developed a novel discrete variable representation (DVR) method where not only the amplitudes of the wave function at the DVR grid points can change but also the positions of these grid points can move as a function of time. Since the Gauss-Hermite basis set is used as the primitive basis functions (PBF) to construct the DVR basis set, the method appears as a semiclassical one with a small number of PBF but converges very fast to the quantum with an increasing PBF. We have investigated the dynamics of a reaction coordinate with or without coupling to a heat bath of harmonic oscillators to demonstrate the validity of the proposed method. The excellent agreement of the calculated tunneling probabilities with numbers obtained by traditional quantum grid method (FFT) and the fast computability of the present method compared to the latter are remarkable. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The collision of a hydrogen molecule with a copper surface has been studied using a semiclassical theory. The four molecular coordinates, the distance from the surface, the vibrational coordinate, and the polar angles, are treated quantum mechanically using a grid method. The kinetic energy is evaluated by a fast Fourier transformation (FFT) technique and the wave function propagated by a Lanczos iterative method. Two translational degrees of freedom for motion at the lattice are treated classically, whereas the normal modes of the solid are quantized using a quantum boson approach. The present calculation indicates that rotational state distributions of the scattered molecule and dissociative chemisorption of the diatom on the metal surface are significantly affected by the phonon coupling. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We calculate the adiabatic Potential Energy Surfaces (PESs) and the Non-Adiabatic Coupling Terms (NACTs) for the three lowest singlet states of H+3 in hyperspherical coordinates as functions of hyperangles (θ and ɸ) for a grid of fixed values of hyperradius (1.5 ≤ ρ ≤ 20 bohrs) using the MRCI level of methodology employing ab initio quantum chemistry package (MOLPRO). The NACT between the ground and the first excited state translates along the seams on the θ - ɸ space, i.e., there are six Conical Intersections (CIs) at each θ (60° ≤ θ ≤ 90°) within the domain, 0 ≤ ɸ ≤ 2p. While transforming the adiabatic PESs to the diabatic ones, such surfaces show up six crossings along those seams. Our beyond Born-Oppenheimer approach could incorporate the effect of NACTs accurately and construct single-valued, continuous, smooth, and symmetric diabatic PESs. Since the location of CIs and the spatial amplitudes of NACTs are most prominent around ρ = 10 bohrs, generally only those results are depicted. [ABSTRACT FROM AUTHOR]

We construct the effective Hartree potential for H2 on Cu(111) as introduced in our earlier work [Dutta et al., J. Chem. Phys. 154, 104103 (2021), and Dutta et al., J. Chem. Phys. 157, 194112 (2022)] starting from the same gas-metal interaction potential obtained for 0 K. Unlike in that work, we now explicitly account for surface expansion at 925 K and investigate different models to describe the surface vibrational modes: (i) a cluster model yielding harmonic normal modes at 0 K and (ii) slab models resulting in phonons at 0 and 925 K according to the quasi-harmonic approximation-all consistently calculated at the density functional theory level with the same exchange-correlation potential. While performing dynamical calculations for the H2(v = 0, j = 0)-Cu(111) system employing Hartree potential constructed with 925 K phonons and surface temperature, (i) the calculated chemisorption probabilities are the highest compared to the other approaches over the energy domain and (ii) the threshold for the reaction probability is the lowest, in close agreement with the experiment. Although the survival probabilities (v' = 0) depict the expected trend (lower in magnitude), the excitation probabilities (v' = 1) display a higher magnitude since the 925 K phonons and surface temperature are more effective for the excitation process compared to the phonons/normal modes obtained from the other approaches investigated to describe the surface., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

The effect of surface mode vibrations on the reactive scattering of D 2 , initialized in the ground rovibrational state (v = 0, j = 0), from a Cu(111) surface is investigated for different surface temperature situations. We adopt a time and temperature dependent effective Hamiltonian [Dutta et al., J. Chem. Phys. 154, 104103 (2021)] constructed by combining the linearly coupled many oscillator model [Sahoo et al., J. Chem. Phys. 136, 084306 (2012)] and the static corrugation model [M. Wijzenbroek and M. F. Somers, J. Chem. Phys. 137, 054703 (2012)] potential within the mean-field approach. Such an effective Hamiltonian is employed for six-dimensional quantum dynamical calculations to obtain temperature dependent reaction and state-to-state scattering probability profiles as a function of incidence energy of colliding D 2 molecules. As reported in the experimental studies, the movements of surface atoms modify the dissociative scattering dynamics at higher surface temperature by exhibiting vibrational quantum and surface atoms' recoil effects in the low and high collision energy domains, respectively. Finally, we compare our present theoretical results with the experimental and other theoretical outcomes, as well as discuss the novelty of our findings.

The effect of surface atom vibrations on H 2 scattering from a Cu(111) surface at different temperatures is being investigated for hydrogen molecules in their rovibrational ground state (v = 0, j = 0). We assume weakly correlated interactions between molecular degrees of freedom and surface modes through a Hartree product type wavefunction. While constructing the six-dimensional effective Hamiltonian, we employ (a) a chemically accurate potential energy surface according to the static corrugation model [M. Wijzenbroek and M. F. Somers, J. Chem. Phys. 137, 054703 (2012)]; (b) normal mode frequencies and displacement vectors calculated with different surface atom interaction potentials within a cluster approximation; and (c) initial state distributions for the vibrational modes according to Bose-Einstein probability factors. We carry out 6D quantum dynamics with the so-constructed effective Hamiltonian and analyze sticking and state-to-state scattering probabilities. The surface atom vibrations affect the chemisorption dynamics. The results show physically meaningful trends for both reaction and scattering probabilities compared to experimental and other theoretical results.

We employ theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation (C 6 H 6 + ) for the first time and explore the workability of the time-dependent discrete variable representation (TDDVR) method for carrying out dynamical calculations to evaluate the photoelectron (PE) spectra of its neutral analog. Ab initio adiabatic PESs and nonadiabatic coupling terms are computed over a series of pairwise normal modes, which exhibit rich nonadiabatic interactions starting from Jahn-Teller interactions and accidental conical intersections/seams to pseudo Jahn-Teller couplings. Once the electronic structure calculation is completed on the low-lying five doublet electronic states (X̃ 2 E 1g , B̃ 2 E 2g , and C̃ 2 A 2u ) of the cationic species, diabatization is carried out employing the adiabatic-to-diabatic transformation (ADT) equations for the five-state sub-Hilbert space to compute highly accurate ADT angles, and thereby, single-valued, smooth, symmetric, and continuous diabatic PESs and couplings are constructed. Subsequently, such surface matrices are used to perform multi-state multi-mode nuclear dynamics for simulating PE spectra of benzene. Our theoretical findings clearly depict that the spectra for X̃ 2 E 1g and B̃ 2 E 2g -C̃ 2 A 2u states obtained from BBO treatment and TDDVR dynamics exhibit reasonably good agreement with the experimental results as well as with the findings of other theoretical approaches.

First principles based beyond Born-Oppenheimer theory has been implemented on the F + H 2 system for constructing multistate global diabatic Potential Energy Surfaces (PESs) through the incorporation of Nonadiabatic Coupling Terms (NACTs) explicitly. The spin-orbit (SO) coupling effect on the collision process of the F + H 2 reaction has been included as a perturbation to the non-relativistic electronic Hamiltonian. Adiabatic PESs and NACTs for the lowest three electronic states (1 2 A', 2 2 A', and 1 2 A″) are determined in hyperspherical coordinates as functions of hyperangles for a grid of fixed values of the hyperradius. Jahn-Teller (JT) type conical intersections between the two A' states translate along C 2v and linear geometries in F + H 2 . In addition, A' and A″ states undergo Renner-Teller (RT) interaction at collinear configurations of this system. Both JT and RT couplings are validated by integrating NACTs along properly chosen contours. Subsequently, we have solved adiabatic-to-diabatic transformation (ADT) equations to evaluate the ADT angles for constructing the diabatic potential matrix of F + H 2 , including the SO coupling terms. The newly calculated diabatic PESs are found to be smooth, single-valued, continuous, and symmetric and can be invoked for performing accurate scattering calculations on the F + H 2 system.

In order to circumvent numerical inaccuracy originating from the singularity of nonadiabatic coupling terms (NACTs), we need to perform kinetically coupled adiabatic to potentially coupled diabatic transformation of the nuclear Schrödinger Equation. Such a transformation is difficult to achieve for higher dimensional sub-Hilbert spaces due to inherent complicacy of adiabatic to diabatic transformation (ADT) equations. Nevertheless, detailed expressions of ADT equations are formulated for six coupled electronic states for the first time and their validity is extensively examined for a well-known radical cation, namely, 1,3,5-C 6 H 3 F 3 + (TFBZ + ). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states (X̃ 2 E '' , Ã 2 A 2 '' , B̃ 2 E ' , and C̃ 2 A 2 ' ), where several types of nonadiabatic interactions, like Jahn-Teller conical intersections (CI), accidental CIs, accidental seams (series of degenerate points), and pseudo Jahn-Teller interactions can be observed over the Franck-Condon region of nuclear configuration space. Those interactions are depicted by exploring degenerate components of C-C asymmetric stretching, C-C symmetric stretching, and C-C-C scissoring motion (Q 9x , Q 9y , Q 10x , Q 10y , Q 12x , and Q 12y ) to compute complete active space self-consistent field level adiabatic PESs and NACTs as implemented in the MOLPRO quantum chemistry package. Subsequently, we perform the ADT using our newly devised fifteen (15) ADT equations to locate the position of CIs, verify the quantization of NACTs, and to construct highly accurate diabatic PESs.

The workability of beyond Born-Oppenheimer theory to construct diabatic potential energy surfaces (PESs) of a charge transfer atom-diatom collision process has been explored by performing scattering calculations to extract accurate integral cross sections (ICSs) and rate constants for comparison with most recent experimental quantities. We calculate non-adiabatic coupling terms among the lowest three singlet states of H 3 + system (1 1 A ' , 2 1 A ' , and 3 1 A ' ) using MRCI level of calculation and solve the adiabatic-diabatic transformation equation to formulate the diabatic Hamiltonian matrix of the same process [S. Mukherjee et al., J. Chem. Phys. 141, 204306 (2014)] for the entire region of nuclear configuration space. The nonadiabatic effects in the D + reaction has been studied by implementing the coupled 3D time-dependent wave packet formalism in hyperspherical coordinates [S. Adhikari and A. J. C. Varandas, Comput. Phys. Commun. 184, 270 (2013)] with zero and non-zero total angular momentum (J) on such newly constructed accurate (ab initio) diabatic PESs of H 2 reaction has been studied by implementing the coupled 3D time-dependent wave packet formalism in hyperspherical coordinates [S. Adhikari and A. J. C. Varandas, Comput. Phys. Commun. 184, 270 (2013)] with zero and non-zero total angular momentum (J) on such newly constructed accurate (ab initio) diabatic PESs of H 3 + . We have depicted the convergence profiles of reaction probabilities for the reactive non-charge transfer, non-reactive charge transfer, and reactive charge transfer processes for different collisional energies with respect to the helicity (K) and total angular momentum (J) quantum numbers. Finally, total and state-to-state ICSs are calculated as a function of collision energy for the initial rovibrational state (v = 0, j = 0) of the H 2 molecule, and consequently, those quantities are compared with previous theoretical and experimental results.

A thorough investigation has been performed for electronic structure, topological effect, and nuclear dynamics of NO2 molecule, where the adiabatic potential energy surfaces (PESs), conical intersections between the ground (X(2)A1) and the first excited state (A(2)B2), and the corresponding non-adiabatic coupling terms between those states are recalculated [Chem. Phys. 416, 11 (2013)] to achieve enough accuracy in dynamics. We employ beyond Born-Oppenheimer theory for these two state sub-Hilbert space to carry out adiabatic to diabatic transformation (ADT) to obtain the ADT angles and thereby, to construct single-valued, smooth, and continuous diabatic PESs. The analytic expressions for the adiabatic PESs and ADT angles are provided to represent a two-state three-mode diabatic Hamiltonian of NO2 for performing nuclear dynamics to calculate the photo-electron spectra of its anion. It appears that not only Jahn-Teller type coupling but also Renner-Teller interaction contributes significantly on the overall spectrum. The coupling between the electronic states (X(2)A1 and A(2)B2) of NO2 is essentially through the asymmetric stretching mode, where the functional form of such interaction is distinctly symmetric and non-linear.

We propose a strategy of using a stochastic optimization technique, namely, simulated annealing to design optimum laser pulses (both IR and UV) to achieve greater fluxes along the two dissociating channels (O(18) + O(16)O(16) and O(16) + O(16)O(18)) in O(16)O(16)O(18) molecule. We show that the integrated fluxes obtained along the targeted dissociating channel is larger with the optimized pulse than with the unoptimized one. The flux ratios are also more impressive with the optimized pulse than with the unoptimized one. We also look at the evolution contours of the wavefunctions along the two channels with time after the actions of both the IR and UV pulses and compare the profiles for unoptimized (initial) and optimized fields for better understanding the results that we achieve. We also report the pulse parameters obtained as well as the final shapes they take.

A recently proposed coupled three-dimensional time-dependent wave-packet formalism in hyperspherical coordinates is shown to yield accurate results for the reactive non-charge transfer process in the title system at collision energies as low as 100 K, where the lowest sheet of the accurate double many body expansion form for the singlet H3 (+) is used. The results are compared with available experimental data as well as time-independent calculations, and the agreement shown to be generally good.

We calculate the adiabatic Potential Energy Surfaces (PESs) and the Non-Adiabatic Coupling Terms (NACTs) for the three lowest singlet states of H3 (+) in hyperspherical coordinates as functions of hyperangles (θ and ϕ) for a grid of fixed values of hyperradius (1.5 ⩽ ρ ⩽ 20 bohrs) using the MRCI level of methodology employing ab initio quantum chemistry package (MOLPRO). The NACT between the ground and the first excited state translates along the seams on the θ - ϕ space, i.e., there are six Conical Intersections (CIs) at each θ (60° ⩽ θ ⩽ 90°) within the domain, 0 ⩽ ϕ ⩽ 2π. While transforming the adiabatic PESs to the diabatic ones, such surfaces show up six crossings along those seams. Our beyond Born-Oppenheimer approach could incorporate the effect of NACTs accurately and construct single-valued, continuous, smooth, and symmetric diabatic PESs. Since the location of CIs and the spatial amplitudes of NACTs are most prominent around ρ = 10 bohrs, generally only those results are depicted.

The tetra-atomic C(2)H(2)(+) cation is known to form Renner-Teller-type intersections along its collinear axis. Not too long ago, we studied the nonadiabatic coupling terms (NACTs) of this molecule [G. J. Halász et al., J. Chem. Phys. 126, 154309 (2007)] and revealed two kinds of intersections. (i) By employing one of the hydrogens as a test particle, we revealed the fact that indeed the corresponding (angular) NACTs produce topological (Berry) phases that are equal to 2pi, which is a result anticipated in the case of Renner-Teller intersections. (ii) However, to our big surprise, repeating this study when one of the atoms (in this case a hydrogen) is shifted from the collinear arrangement yields for the corresponding topological phase a value that equals pi (and not 2pi). In other words, shifting (even slightly) one of the atoms from the collinear arrangement causes the intersection to change its character and become a Jahn-Teller intersection. This somewhat unexpected novel result was later further analyzed and confirmed by other groups [e.g., T. Vertesi and R. Englman, J. Phys. B 41, 025102 (2008)]. The present article is devoted to another tetra-atomic molecule, namely, the H(2)CN molecule, which just like the C(2)H(2)(+) ion, is characterized by Renner-Teller intersections along its collinear axis. Indeed, we revealed the existence of Renner-Teller intersections along the collinear axis, but in contrast to the C(2)H(2)(+) case a shift of one atom from the collinear arrangement did not form Jahn-Teller intersections. What we found instead is that the noncollinear molecule was not affected by the shift and kept its Renner-Teller character. Another issue treated in this article is the extension of (the two-state) Berry (topological) phase to situations with numerous strongly interacting states. So far the relevance of the Berry phase was tested for systems characterized by two isolated interacting states, although it is defined for any number of interacting states [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. We intend to show how to overcome this limitation and get a valid, fully justified definition of a Berry phase for an isolated system of any number of interacting states (as is expected).

When a set of three states is coupled with each other but shows negligibly weak interaction with other states of the Hilbert space, these states form a sub-Hilbert space. In case of such subspace [J. Chem. Phys. 124, 074101 (2006)], (a) the adiabatic-diabatic transformation (ADT) condition, nablaA + tauA = 0 [Chem. Phys. Lett. 35, 112 (1975)], provides the explicit forms of the nonadiabatic coupling (NAC) elements in terms of electronic basis function angles, namely, the ADT angles, and (b) those NAC terms satisfy the so-called curl conditions [Chem. Phys. Lett. 35, 112 (1975)], which ensure the removal of the NAC elements [could be singular also at specific point(s) or along a seam in the configuration space] during the ADT to bring the diabatic representation of the nuclear Schrodinger equation with a smooth functional form of coupling elements among the electronic states. Since the diabatic to adiabatic representation of the Hamiltonian is related through the same unitary transformation (nablaA + tauA = 0), it could be quite interesting to explore the nature of the nonadiabatic coupling terms starting from a diabatic Hamiltonian and, thereafter, to formulate the extended Born-Oppenheimer (EBO) equation for those adiabatic states transformed from diabatic ones. We consider a three-state diabatic potential matrix constructed for the excited states of Na(3) cluster [J. Chem. Phys. 88, 6068 (1988)] at the pseudo-Jahn-Teller model situation, which can reproduce experimentally measured vibrationally resolved absorption lines [Surf. Sci. 156, 770 (1985)] with appropriate choice of coupling parameters, analytically calculate the nonadiabatic coupling elements along with their curls, and numerically evaluate the ADT angles to explore the nature of its nonadiabaticity. While formulating the single surface beyond the BO equation, our theoretical derivation demonstrates that the existence of zero curls of the NAC terms is a necessity. Indeed, when the energy gap between the third state (1(2) A(1)(')/2(2) A(1)(')) and the doubly degenerate states (2(2) E(')/3(2) E(')) of the model Hamiltonian for Na(3) cluster is considered to be either identically or approximately zero, the curl for each NAC element naturally approaches zero, leading to a theoretically valid EBO equation. We demonstrate the numerical validity of the EBO equation by calculating the nonadiabatic effects on the photoabsorption spectrum starting with the initial wave function located on the ground electronic state and compare with the corresponding diabatic spectrum when the three states are either degenerate at a point or approaching to form three-state degeneracy at the same point. Finally, we calculate the vibrational eigenspectrum of the ground adiabatic state by using (so to say) theoretically and numerically valid EBO equation to compare with those experimentally measured and BO/geometric phase calculated spectra (Tables I-III).