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Energies and derivative couplings in the vicinity of a conical intersection 3. The 'most' diabatic basis

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Abstract

It is shown that in the immediate vicinity of an arbitrary conical intersection at Rx all the derivative coupling, except for the small part due to the finiteness of the basis sets, is removable by the orthogonal transformation generated by the angle α(ρ, θ, z) = λ(θ) /2 + ρmρ(θ) /q(θ) + zmz(θ) /q(θ), where ρ,θ,z are cylindrical polar coordinates centred at Rx. Expressions for λ(θ), q(θ) mρ(θ) and mz(θ) are given. The implications of this result for numerical studies that (i) determine the 'most' diabatic basis using Poisson's equation and (ii) assess approximate diabatization schemes are discussed.

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... At the close proximity of a degeneracy, only the removable coupling is singular and according to the degenerate perturbation theory, the nonremovable coupling is insignificant. 22 It means that the ADT angle can be obtained by integrating the derivative coupling at and around the same region. 19 On the contrary, away from the conical intersection ͑CI͒, the contribution from the nonremovable coupling appears in path-dependent integrals for the ADT angles and therefore, closed line integrals of the derivative coupling 23 will not be multiples of . ...
... Since the vector field due to NAC term could show singularity ͑pole͒ in the configuration space and decays like 1 / r ͑where r is the radial coordinate͒, such field may be resolved 22,23 into longitudinal and transverse components, where the curl of irrotational part is known to be zero but the same for solenoidal component may or may not. When the adiabatic PESs are well separated, the removal and nonremoval ͑with respect to ADT͒ NAC terms appear as of the same order. ...
... On the other hand, at the close proximity of a degeneracy the removal couplings are singularly large and the nonremoval couplings are negligibly small. 22 Moreover, one can evaluate the ADT angles by integrating the derivative couplings ͑removal͒ at and around CIs, but the same ADT angle does not show gauge invariance due to nonremoval components arising while away from the CI. 23 However, it is well accepted 24 that this problem due to nonremoval component can be reduced with the inclusion of more electronic states or automatically subsides with a few numbers of states forming a sub-Hilbert space. ...
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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).
... At sufficiently low energies (well below the energy of the upper state), these coupling can be ignored in dynamics calculations due to the 1/M prefactor. At the close proximity of a degeneracy, only the removable coupling is singular and according to the degenerate perturbation theory, the nonremovable couplings are insignificant [9]. It means that the ADT angle can be obtained by integrating the derivative coupling at and around the same region. ...
... Indeed, it is important to note that each ADT matrix can provide similar set of differential equations for ADT angles [Eq. (9)], NAC elements [Eq. (10)] and their Curl-Divergence equations [Eqs. ...
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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
... The non-adiabatic coupling terms (NACTs) could play an important role to understand the mechanism of spectroscopic and scattering problems with or without radiation-less charge transfer processes. One can handle those terms elegantly [1,456789101112131718192021 to construct the so called ''diabatic'' (potentially coupled) Hamiltonian matrix instead of neglecting them forcefully for any arbitrary situation of interaction among the electronic states. Since the adiabatic potential energy surfaces (PESs) and the NACTs are unique and physically realizable, any formal development and its numerical implementation on beyond Born–Oppenheimer (BO) theory may naturally expected to start form the same representation (adiabatic PESs and NACTs). ...
... As the removable components of the NACTs are usually sharp functions of nuclear coordinates due to the singularities located at anywhere in the configuration space (CS), it is necessary to perform an unitary transformation from the kinetically coupled (adiabatic) Hamiltonian to the potentially (diabatic) one, where all the terms of the Hamiltonian would be continuous and smooth functions of nuclear coordinates and thereby, dynamical calculations could be accurate and stable. The transformation from adiabatic to diabatic representation of SE for a given sub-Hilbert space is guaranteed to produce ''correct'' and ''accurate'' diabatic PESs under the following conditions: (a) The non-removable components of the NACTs are approximately zero [10]; (b) The vector fields created by the NACTs satisfy the so called Curl Conditions [8,18]; (c) The ab initio calculated NACTs are adapted [17,21] with molecular symmetry [22,23] (MS) to assign their appropriate Irreducible Representations (IREPs) so that the nuclear Hamiltonian is totally symmetric under the operation of each element of the corresponding MS group of a molecule. In other words, when the diabatic PESs are continuous and single valued, the existence of a quantization rule [24] for removable components of the NACTs with singularities is predicted for a given sub-Hilbert space and such rule was confirmed [25] subsequently. ...
... There are several methods for the computation of reaction probabilities based on combining Newtonian or Brownian molecular dynamics with adiabatic and quasi-diabatic potential energy surfaces [31][32][33][34][35][36][37][38][39][40][41][42][43]. In these schemes, surface hopping and residence is controlled by the NACTs, in particular near conical intersections. ...
... Despite the different rationale for the occurrence of a quantum-state transition, an algorithm for computing the ERPs can profit from the techniques used elsewhere for exploring the configurational space [31][32][33][34][35][36][37][38][39][40][41][42][43]. A simple implementation would be as follows (cf. ...
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We have recently proposed an approach where chemical transformations can be described as quantum processes involving the modulation of entangled states by an applied external field (Arteca and Tapia in Phys Rev A 84:012115, 2011). In practical implementations, we gain insight into these processes by using simple quantum-mechanical models derived from diabatic schemes. In this context, reactant, product, and, eventually, intermediate species, are assigned to diabatic basis functions, and then entangled by an external field into a quantum state from which all observable properties of the chemical reaction should emerge. Here, we extend our previous model for bond breaking/formation in diatomic molecules (Arteca et al. in J Math Chem 50:949, 2012). We consider the entire manifold of semiclassical models defined by only two diabatic basis functions: a harmonic well for the “molecular” bound state, and an exponential potential energy function for the asymptotically separated fragments (the “product” channel). Using a two-parameter space to describe all models, we determine how the topology of the total energy function is affected by the shape of the applied field. We show that strong and weak local couplings with the external field modify substantially the occurrence of energy barriers, in contrast to using the uniform (i.e., space-invariant) coupling employed in previous works.
... Over the years, a number of formulations of approximate or quasidiabatic or " locally rigorous " diabatic states[20,[25][26][27][28][29][30][31][32][33]have appeared. Only very recently[34][35][36][37][38][39]have there been attempts to use high quality ab initio wave functions to consider the magnitude of the non-removable part of the first-derivative coupling vector. In one such attempt[38], a quasi-diabatic basis was reported for the HeH 2 system by solving a two-dimensional Poisson equation on the plane in three-dimensional configuration space passing through the conical intersection configuration of smallest energy. ...
... The lowest value used for θ was 0.1 @BULLET because the Poisson equation solver in MUDPACK library for spherical polar coordinates is unstable for values of θ below that value. Besides, at such small values of θ, both β and γ are known since the DMBE representation[47](Eqs. (36) and(37)) is quite accurate in these regions. Let us consider the internal configuration space frame OX λ Y λ Z ofFig. ...
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Molecular reaction dynamics in the adiabatic representation is complicated by the existence of conical intersections and the associated geometric phase effect. The first-derivative coupling vector between the corresponding electronically adiabatic states can, in general, be decomposed into longitudinal (removable) and transverse (nonremovable) parts. At intersection geometries, the longitudinal part is singular, whereas the transverse part is not. In a two-electronic-state Born-Huang expansion, an adiabatic-to-diabatic transformation completely eliminates the contribution of the longitudinal part to the nuclear motion Schrödinger equation, leaving however the transverse part contribution. We report here the results of an accurate calculation of this transverse part for the 1 2A' and 2 2A' electronic states of H3 obtained by solving a three-dimensional Poisson equation over the entire domain U of internal nuclear configuration space Q of importance to reactive scattering. In addition to requiring a knowledge of the first-derivative coupling vector everywhere in U, the solution depends on an arbitrary choice of boundary conditions. These have been picked so as to minimize the average value over U of the magnitude of the transverse part, resulting in an optimal diabatization angle. The dynamical importance of the transverse term in the diabatic nuclear motion Schrödinger equation is discussed on the basis of its magnitude not only in the vicinity of the conical intersection, but also over all of the energetically accessible regions of the full U domain. We also present and discuss the diabatic potential energy surfaces obtained by this optimal diabatization procedure.
... Since the non -adiabatic coupling terms (NACTs) have important roles [1][2][3][4][5][6][7][8][9][10][11][12] in the description of scattering and radiation -less processes in molecular system, it is a matter of contemporary research how gracefully one can handle the NACTs instead of neglecting them forcibly [13][14][15][16][17][18][19][20] . As the adiabatic potential energy surfaces (PESs) and the NACTs are physically indicative and the removable components of those NACTs are usually very sharp functions of nuclear coordinates with singularities 21,22 in the configuration space (CS), one needs to perform a unitary transformation to obtain the diabatic representation, where couplings among the electronic states are smooth functions of nuclear coordinates and thereby, the dynamical calculations on the diabatic PESs would be numerically accurate and stable. ...
... Since we need to select the correct combination from the above two possible ones, we utilize the following quantization rule 17 [Eq. (15)] of the contour integrals over the NACTs evaluated along a closed loop L g of nuclear coordinate s n around a single conical intersection (CI): ...
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We calculate the adiabatic Potential Energy Surfaces (PESs) and the Non - Adiabatic Coupling Terms (NACTs) for the excited electronic states (22 E' and 12 A'1) of Na3 cluster at the MRCI level by using ab initio quantum chemistry package (MOLPRO), where the NACTs are adapted with Molecular Symmetry (MS) by employing appropriate Irreducible Representations (IREPs). Such terms are incorporated into the Adiabatic to Diabatic Transformation (ADT) equations to obtain the ADT angles to construct the continuous, single - valued, symmetric and smooth 3 × 3 diabatic Hamiltonian matrix.
... At the close vicinity of a degenerate point in nuclear configuration space (CS), the removable one steeply rises to infinity, whereas the nonremovable component appears to be insignificant. 21 Nevertheless, adiabatic to diabatic transformation (ADT) 22,23 can only be able to eliminate the contribution of the longitudinal component, expressed as the derivative of a scalar. 19,21 According to Cauchy's residue theorem, the ADT angles attain the magnitude of the integer multiple of π at the end of a closed contour 24,25 encircling any degenerate point(s), but those mixing angles fail to acquire the desired magnitude, if the non-removable component has a significant contribution at the close vicinity of the conical intersection (CI) points. ...
... 21 Nevertheless, adiabatic to diabatic transformation (ADT) 22,23 can only be able to eliminate the contribution of the longitudinal component, expressed as the derivative of a scalar. 19,21 According to Cauchy's residue theorem, the ADT angles attain the magnitude of the integer multiple of π at the end of a closed contour 24,25 encircling any degenerate point(s), but those mixing angles fail to acquire the desired magnitude, if the non-removable component has a significant contribution at the close vicinity of the conical intersection (CI) points. 26 The magnitude of the non-removable component can be minimized by including a higher number of electronic states in the interested domain of nuclear CS. ...
Article
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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-C6H3F3+ (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states (X̃2E'', Ã2A2'', B̃2E', and C̃2A2'), 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.
... 29,30 Also, it has been observed that at the close proximity of a degeneracy, the removable part of the coupling blows up, but the non-removable component remains insignificant. 31 The ADT angle(s) can be computed by integrating the derivative coupling at and around the CI point(s) or seam(s), and the close line integrals of such couplings will result into nπ, n being any integer. 28,32,33 Baer's approach involving line integral over the NACTs to transform the adiabatic SE to diabatic framework for two-electronic state sub-Hilbert space (SHS) has been successful. ...
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We review the formulation of beyond Born-Oppenheimer (BBO) theory based on first principle for the construction of diabatic potential energy surfaces (PESs) both for few important spectroscopic systems, viz., Na 3 cluster, NO 2 radical as well as scattering process like D + + H 2 . The essential theoretical development leading to the BBO equations are thoroughly discussed. It has been found that the above molecular systems posses numerous nonadiabatic interactions that range from Jahn-Teller, Renner-Teller types of conical intersections along with strong pseudo Jahn-Teller couplings between various electronic states. We have calculated the adiabatic PESs and nonadiabatic coupling terms for those systems and subsequently performed adiabatic-to-diabatic transformation to construct smooth, symmetric and continuous diabatic potential energy matrix. Nuclear dynamics has been performed on the diabatic PESs of Na 3 and NO 2 to simulate the photoelectron spectra that match quite well with the experimentally measured ones. Moreover, we have carried out reactive scattering dynamics on the adiabatic and diabatic surfaces of H + 3 system to reproduce experimental cross sections for reactive charge and non-charge transfer processes.
... 29,30 Also, it has been observed that at the close proximity of a degeneracy, the removable part of the coupling blows up, but the non-removable component remains insignificant. 31 The ADT angle(s) can be computed by integrating the derivative coupling at and around the CI point(s) or seam(s), and the close line integrals of such couplings will result into nπ, n being any integer. 28,32,33 Baer's approach involving line integral over the NACTs to transform the adiabatic SE to diabatic framework for two-electronic state sub-Hilbert space (SHS) has been successful. ...
Article
Full-text available
We review the formulation of beyond Born-Oppenheimer (BBO) theory based on first principle for the construction of diabatic potential energy surfaces (PESs) both for few important spectroscopic systems, viz., Na3 cluster, NO2 radical as well as scattering process like D⁺ + H2. The essential theoretical development leading to the BBO equations are thoroughly discussed. It has been found that the above molecular systems posses numerous nonadiabatic interactions that range from Jahn-Teller, Renner-Teller types of conical intersections along with strong pseudo Jahn-Teller couplings between various electronic states. We have calculated the adiabatic PESs and nonadiabatic coupling terms for those systems and subsequently performed adiabatic-to-diabatic transformation to construct smooth, symmetric and continuous diabatic potential energy matrix. Nuclear dynamics has been performed on the diabatic PESs of Na3 and NO2 to simulate the photoelectron spectra that match quite well with the experimentally measured ones. Moreover, we have carried out reactive scattering dynamics on the adiabatic and diabatic surfaces of system to reproduce experimental cross sections for reactive charge and non-charge transfer processes.
... tion of the electronic basis. However, near CIs, the regions where accuracy of the diabatisation procedure is most important, the nonremovable couplings (to the excluded states) are insignificant[38] and the expression valid. ...
Article
An extension of a recent diabatisation scheme for use in direct-dynamics variational multi-configuration Gaussian (DD-vMCG) quantum dynamics calculations is presented which allows the treatment of systems with more than two electronic states. Methodological updates to the DD-vMCG implementation are presented along with applications of the method to 2-, 3- and 4-state models of the butatriene cation. As a demonstration of the utility of the method, results of 3-state, full-dimensional calculations on the DNA base, thymine, are included, showing the energy dissipation through wavefunction population transfer between states.
... It means that each diabatic matrix element should be a fairly simple function of the nuclear coordinates and thus is comparably easy to model. Explicit approaches for the diabatization of adiabatic wave functions and energies can be classified in two groups, approaches calculating and annihilating the derivative couplings [42][43][44][45][46][47][48][49] and methods avoiding the explicit calculation of the derivative couplings. The latter group can be subdivided into approaches that do 38,40,[50][51][52][53][54][55][56][57][58][59][60] or do not 34,61-67 require an explicit wave function to find the adiabatic-diabatic transformation. ...
Article
Robust diabatization techniques are key for the development of high-dimensional coupled potential energy surfaces (PESs) to be used in multi-state quantum dynamics simulations. In the present study we demonstrate that, besides the actual diabatization technique, common problems with the underlying electronic structure calculations can be the reason why a diabatization fails. After giving a short review of the theoretical background of diabatization, we propose a method based on the block-diagonalization to analyse the electronic structure data. This analysis tool can be used in three different ways: First, it allows to detect issues with the ab initio reference data and is used to optimize the setup of the electronic structure calculations. Second, the data from the block-diagonalization are utilized for the development of optimal parametrized diabatic model matrices by identifying the most significant couplings. Third, the block-diagonalization data are used to fit the parameters of the diabatic model, which yields an optimal initial guess for the non-linear fitting required by standard or more advanced energy based diabatization methods. The new approach is demonstrated by the diabatization of 9 electronic states of the propargyl radical, yielding fully coupled full-dimensional (12D) PESs in closed form.
... They showed that this kind of transformation can at best remove the longitudinal component of NACTs. In later works, Yarkony and co-workers observed that at close proximity of a CI, the removable part of the derivative coupling vector (longitudinal component) blows up, but the nonremovable component (transverse) remains insignificant [39]. The general prescription is that in order to compute the transformation angle(s), the NACTs need to be integrated at and around the CI point(s) or seam(s), where their close contour integrals will result into nπ with n being an integer [40][41][42]. ...
Article
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We review our development on beyond Born-Oppenheimer (BBO) theory and its implementation on various models and realistic molecular processes as carried out over the last 15 years. The theoretical formulation leading to the BBO equations are thoroughly discussed with ab initio calculations. We have employed first principle based BBO theory not only to formulate single surface extended Born-Oppenheimer equation to understand the nature of nonadiabatic effect but also to construct accurate diabatic potential energy surfaces (PESs) for important spectroscopic systems, namely, NO 2 radical, Na 3 and K 3 clusters, NO 3 radical, benzene and 1,3,5-trifluorobenzene radical cations (C 6 H + 6 and C 6 H 3 F + 3) as well as triatomic reactive scattering systems like H + + H 2 and F + H 2. The nonadiabatic phenomena like Jahn-Teller (JT), Renner-Teller, pseudo Jahn-Teller effects and the accidental conical intersections are the key players in dictating spectroscopic and reactive scattering profiles. The nature of diabatic coupling elements derived from ab initio data with BBO theory for molecular processes in Franck-Condon region has been analysed in the context of linearly and bilinearly coupled JT model Hamiltonian. The results obtained from quantum dynamical calculations on BBO based diabatic PESs of the above molecular systems are found to be in accord with available experimental outcomes.
... If the required number of electronic states forming the sub-space are incorporated in ADT calculation, the computed mixing (ADT) angles result in an integer (n) multiple of p 39,41,42 along a closed loop enclosing degenerate point(s) or passing through a seam. On the other hand, Mead et al. 28,43 predicted that ADT can at best eliminate the longitudinal component (removable part) of NACTs, which blows up in the vicinity of a conical intersection (CI) or seam, 44 but the magnitude of the transverse component (non-removable part) is generally negligible at those regions or can be made negligible by expanding the nuclear CS and/or the sub-Hilbert space. Subsequently, various diabatization methods have been proposed (see Subsection 6.1 for details), namely vibronic coupling model (VCM), [45][46][47][48][49][50][51][52][53][54][55][56][57][58] exact factorisation (EF) [59][60][61] scheme, direct dynamical approaches [62][63][64][65][66] and also other methods, [67][68][69][70] but the first principle based approach developed by M. Baer 3,[38][39][40][41][42] is considered as one of the most accurate ones. ...
Article
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We present first principle based beyond Born-Oppenheimer (BBO) theory and its applications on various models as well as realistic spectroscopic and scattering processes, where the Jahn-Teller (JT) theory is brought in conjunction with BBO approach on the phase transition of lanthanide complexes. Over one and half decades, our development of BBO theory is demonstrated with ab initio calculations on representative molecules of spectroscopic interest (NO2 radical, Na3 and K3 clusters, NO3 radical, C6H6+ and 1,3,5-C6H3F3+ radical cations) as well as triatomic reactive scattering processes (H++H2 and F+H2). Such an approach exhibits the effect of JT, Renner-Teller (RT) and pseudo Jahn-Teller (PJT) type of interactions. While implementing the BBO theory, we generate highly accurate diabatic potential energy surfaces (PESs) to carry out quantum dynamics calculation and find excellent agreement with experimental photoelectron spectra of spectroscopic systems and cross-sections/rate constants of scattering processes. On the other hand, such electron-nuclear couplings incorporated through JT theory play the crucial role in dictating higher energy satellite transitions in the dielectric function spectra of LaMnO3 complex. Overall, this article thoroughly sketch the current perspective of BBO approach and its connection with JT theory with various applications on physical and chemical processes.
... I n recent years, a lot of efforts were invested in studying the nature of the electronic nonadiabatic coupling terms (NACTs)1234567891011. The NACTs are characterized by two features: They are vectors (in contrast to potentials, which are scalars) and they may become singular (in contrast to potentials, which do not). ...
Article
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
Article
We revisit the notion of structural similarity along a reaction path within the context of a generalized electronic diabatic (GED) molecular model. In this approach, a reaction involving two closed-shell stable species is described as the evolution of a quantum state that superimposes at least three diabatic electronic species (reactant, product, and an open-shell transition state) coupled by an external electromagnetic field. Reactant and product amplitudes in this general state are also modulated by changing the geometry of a system of classical positive charges interacting with the electrons. By mapping these amplitudes over nuclear configurational space, we can follow the total quantum state along a reaction coordinate and establish its similarity to each of the diabatic species. As a result, chemical processes, and useful notions such as those of energy barriers and the Hammond postulate, emerge as consequence of Franck–Condon-like transitions between quantum states. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
Article
An “equation of motion” for matrix elements of an arbitrary Hermitian operator with respect to nuclear coordinates is derived. In the diabatic basis, this equation expresses the smoothness of the corresponding molecular property. Its solution, which determines an adiabatic-to-diabatic transformation, is considered in the two- and three-state approximations. The relation between a smoothness of molecular property and a configurational uniformity introduced by Atchity and Ruedenberg is discussed. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 235–243, 2000
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Active site properties in some proteins can be affected by conformational fluctuations of neighbor residues, even when the latter are not involved directly in the binding process. A local environment thus appears to alter the relevant potential energy surface and its reaction paths. Here, some aspects of this phenomenon are simulated within a generalized electronic diabatic (GED) scheme to study the geometry and structural similarity for a class of two-dimensional (2D) energy surfaces. The electronic quantum state is a linear superposition of diabatic basis functions, each of which is taken to represent a single (pure) electronic state for the isolated material system. Here, we describe a model reaction of isomerization by shifts in amplitudes for three diabatic species (reactant, product, and an open-shell transition state) coupled in an external field. The “effective” 2D energy surface in the field is characterized in terms of critical points, and the amplitudes along the main reaction paths. A new feature is the introduction of a phase diagram where all possible potential-energy-surface topologies (consistent with three-state systems in two linear coordinates) are matched with actual model parameters. By varying the coupling strengths between diabatic states, we classify regions of this phase diagram in terms of electronic and structural similarities; some regions comprise models whose reaction paths have geometries that belong to the catchment region of the reactant, yet are electronically akin to the diabatic transition state or product. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
Chapter
Overview of Reactions Requiring Two States Spin-Forbidden Reaction, Intersystem CrossingSpin–Orbit Coupling as a Mechanism for Spin-Forbidden Reactions General ConsiderationsAtomic Spin–Orbit CouplingMolecular Spin–Orbit CouplingCrossing ProbabilityFermi Golden RuleLandau–Zener Semiclassical ApproximationMethodology for Obtaining Spin–Orbit Matrix Elements Electron Spin in Non-Relativistic Quantum MechanicsKlein–Gordon EquationDirac EquationFoldy–Wouthuysen TransformationBreit–Pauli HamiltonianZeff MethodEffective Core Potential-Based MethodModel Core Potential-Based MethodDouglas–Kroll TransformationPotential Energy Surfaces Minimum Energy Crossing Point LocationAvailable Programs for Modeling Spin-Forbidden ReactionsApplications to Spin-Forbidden Reactions Diatomic MoleculesPolyatomic MoleculesMolecular PropertiesDynamical AspectsOther ReactionsBiological Chemistry Spin-Forbidden Reaction, Intersystem Crossing General ConsiderationsAtomic Spin–Orbit CouplingMolecular Spin–Orbit CouplingCrossing ProbabilityFermi Golden RuleLandau–Zener Semiclassical Approximation Electron Spin in Non-Relativistic Quantum MechanicsKlein–Gordon EquationDirac EquationFoldy–Wouthuysen TransformationBreit–Pauli HamiltonianZeff MethodEffective Core Potential-Based MethodModel Core Potential-Based Method Douglas–Kroll Transformation Minimum Energy Crossing Point Location Diatomic MoleculesPolyatomic MoleculesMolecular PropertiesDynamical Aspects
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The method discussed in this work provides a theoretical framework where simple chemical reactions resemble any other standard quantum process, i.e., a transition in quantum state mediated by the electromagnetic field. In our approach, quantum states are represented as a superposition of electronic diabatic basis functions, whose amplitudes can be modulated by the field and by the external control of nuclear configurations. Using a one-dimensional three-state model system, we show how chemical structure and dynamics can be represented in terms of these control parameters, and propose an algorithm to compute the reaction probabilities. Our analysis of effective energy barriers generalizes previous ideas on structural similarity between reactant, and product, and transition states using the geometry of conventional reaction paths. In the present context, exceptions to empirical rules such as the Hammond postulate appear as effects induced by the environment that supplies the external field acting on the quantum system.
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We have recently shown how high-accuracy wave function grid-based propagation schemes, such as the multiconfiguration time-dependent Hartree (MCTDH) method, can be combined with machine-learning (ML) descriptions of PESs to yield an "on-the-fly" direct dynamics scheme which circumvents potential energy surface (PES) prefitting. To date, our approach has been demonstrated in the ground-state dynamics and nonadiabatic spin-allowed dynamics of several molecular systems. Expanding on this successful previous work, this Article demonstrates how our ML-based quantum dynamics scheme can be adapted to model nonadiabatic dynamics for spin-forbidden processes such as intersystem crossing (ISC), opening up new possibilities for modeling chemical dynamic phenomena driven by spin-orbit coupling. After describing modifications to diabatization schemes to enable accurate and robust treatment or electronic states of different spin-multiplicity, we demonstrate our methodology in applications to modeling ISC in SO2 and thioformaldehyde, benchmarking our results against previous trajectory- and grid-based calculations. As a relatively efficient tool for modeling spin-forbidden nonadiabatic dynamics without demanding any prefitting of PESs, our overall strategy is a potentially powerful tool for modeling important photochemical systems, such as photoactivated pro-drugs and organometallic catalysts.
Article
Complete basis states (BSs), in abstract configuration space-projected quantum mechanics (QM), permit representations of any physical and chemical process elicited by quantum states changes. For a material 1-system, defined by n-electrons and m-nuclei, BSs including relevant fragments cover a representation of chemical species identifiable by spectral response toward electromagnetic (EM) radiations. Reactants, products, and intermediate species are expressed as specific linear superpositions where the amplitude in square modulus at a given BS controls the relative intensity to the spectrum rooted at the corresponding energy eigenstate. Moreover, there is no trace that quantum numbers characterizing BSs would be changed as a function of particular regions of nuclear or electronic configuration space.
Article
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.
Article
We introduce a protocol to represent quantum states as a linear superposition of model electronic diabatic basis states coupled in an external (static) elec. field. By considering an entire family of these models, we uncover trends in reaction-path geometry and the topol. of potential-energy surfaces, including all those that can be realized in a two-dimensional configurational space. Our approach can be used as a tool to model the key parameters (e.g., diabatic basis states, external field intensity) that yield desired geometrical characteristics in an actual potential energy surface. In this work, external agents such as laser fields, or a group of neighboring charges, are regarded as essential requirements to prompt, or trigger, the occurrence of a chem. process. In these cases, reaction path geometry can be modulated externally so as to yield processes that would appear to occur far from gas-phase geometries. This phenomenol. is intrinsically nonadiabatic. Our present approach accounts for the possibility of such features, i.e., the occurrence of quantum states whose electronic structures resemble products, while at geometries that are more similar to those of reactants. Conference Information: 33rd Congress of Theoretical Chemists of Latin ExpressionUniv Havana, Havana, CUBA, SEP 17-21, 2007
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The potential energy surfaces of low-lying states in rhenium tetrahydride (ReH(4)) were explored by using the multiconfiguration self-consistent field (MCSCF) method together with the SBKJC effective core potentials and the associated basis sets augmented by a set of f functions on rhenium atom and by a set of p functions on hydrogen atoms, followed by spin-orbit coupling (SOC) calculations to incorporate nonscalar relativistic effects. The most stable structure of ReH(4) was found to have a D(2d) symmetry and its ground state is (4)A(2). It is found that this is lower in energy than the dissociation limit, ReH(2)+H(2), after dynamic correlation effects are taken into account by using second-order multireference Møller-Plesset perturbation (MRMP2) calculations. This reasonably agrees with previous results reported by Andrews et al. [J. Phys. Chem. 107, 4081 (2003)]. The present investigation further revealed that the dissociation reaction of ReH(4) cannot occur without electronic transition from the lowest quartet state to the lowest sextet state. This spin-forbidden transition can easily occur because of large SOC effects among low-lying states in such heavy metal-containing compounds. The minimum-energy crossing (MEX) point between the lowest quartet and sextet states is proved to be energetically and geometrically close to the transition state for the dissociation reaction on the potential energy surface of the lowest spin-mixed state. The MEX point (C(2) symmetry) was estimated to be 9184 cm(-1) (26.3 kcal/mol) higher than the (4)A(2) state in D(2d) symmetry at the MRMP2 level of theory. After inclusion of SOC effects, an energy maximum on the lowest spin-mixed state appears near the MEX point and is recognized as the transition state for the dissociation reaction to ReH(2)+H(2). The energy barrier for the dissociation, evaluated to be MEX in the adiabatic picture, was calculated to be 5643 cm(-1) (16.1 kcal/mol) on the lowest spin-mixed state when SOC effects were estimated at the MCSCF level of theory.
Article
During the past decade the perception of conical intersections has changed. It is now appreciated that what was once viewed largely as a theoretical curiosity is an essential aspect of electronically nonadiabatic processes. Concomitantly, our understanding of this singular consequence of the Born−Oppenheimer separation of nuclear and electronic motion has grown enormously. In this work the theory of conical intersections is reviewed.
Article
A conical intersection exists between the ground (1 2 A[prime]) and the first-excited (2 2A[prime]) electronic potential energy surfaces (PESs) of the H3 system for C3v geometries. This intersection induces a geometric phase effect, an important factor in accurate quantum mechanical reactive scattering calculations, which at low energies can be performed using the ground PES only, together with appropriate nuclear motion boundary conditions. At higher energies, however, such calculations require the inclusion of both the 1 2A[prime] and 2 2A[prime] electronic PESs and the corresponding nuclear derivative couplings. Here we present ab initio first-derivative couplings for these states obtained by analytic gradient techniques and a fit to these results. We also present a fit to the corresponding 1 2A[prime] and 2 2A[prime] adiabatic electronic PESs, obtained from the ab initio electronic energies. The first-derivative couplings are compared with their approximate analytical counterparts obtained by Varandas et al. [J. Chem. Phys. 86, 6258 (1987)] using the double many-body expansion method. As expected, the latter are accurate close to conical intersection configurations but not elsewhere. We also present the contour integrals of the ab initio couplings along closed loops around the above-mentioned conical intersection, which contain information about possible interactions between the 2 2A[prime] and 3 2A[prime] states.
Article
When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions--the necessary requirements to ensure the adiabatic-diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born-Oppenheimer (EBO) equations [J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.
Article
A method for constructing diabatic potential-energy matrices from ab initio quantum chemistry data is described and tested for use in exact quantum reactive scattering. The method is a refinement of that presented in a previous paper, in that it accounts for the presence of the nonremovable derivative coupling. The accuracy of quantum dynamics on this type of diabatic potential is tested by comparison with an analytic model and for an ab initio description of the two lowest-energy states of H-3. (c) 2005 American Institute of Physics.
Article
This account discusses recent advances in our understanding of conical intersections. Of particular interest is the role of same-symmetry conical intersections, a class of conical intersections of emerging importance. The existence of same-symmetry conical intersections was once a matter of considerable debate. However, as a result of algorithms that locate conical intersections without prior determination of the potential energy surfaces in question, the existence and importance of this class of conical intersections is now firmly established. Here a new role for this class of conical intersections is emphasized. It is observed that symmetry-allowed conical intersections, intersections readily anticipated owing to the role played by point group symmmetry, need not be isolated features. Rather symmetry-allowed and same-symmetry conical intersections can coexist in the same region of nuclear coordinate space and can in fact intersect. A procedure to anticipate these "doubly diabolical points" based only on the knowledge of the symmetry-allowed intersection is reviewed. Doubly diabolical points are potentially quite important. Their existence means that a symmetry-allowed seam of conical intersection may not provide the complete description of nonadiabatic effects in a particular region of nuclear coordinate space. Determining their prevalence in triatomic and general polyatomic molecules will be an important area of future research. Also discussed in this work is a perturbative description of the wave functions near a conical intersection. This analysis provides a transformation to a locally diabatic basis and should facilitate representation of the ab initio potential energy, and derivative couplings, surfaces that exhibit conical intersections.
Article
A Hamiltonian, Hd,(2), “rigorously” diabatic in the vicinity of Rx, a point of conical intersection, is constructed using second-order degenerate perturbation theory. Near an Rx on a C2v seam of conical intersection of two states of different symmetry, Hd,(2) may exhibit a confluence with a Cs seam of conical intersection of two states of the same symmetry. Thus by construction, there exists a “rigorous” diabatic representation of the vicinity of this confluence. A procedure for defining a unique linear combination of the degenerate states at a conical intersection is found to be useful for determining the parameters for Hd,(2) and for identifying approximate symmetries in situations where point group symmetry is rigorously absent. © 2000 American Institute of Physics.
Article
In this article, two issues related to the size of the electronic diabatic potential energy matrix are treated. (a) We frequently mention the fact that the dimension of a diabatic matrix obtained by a unitary transformation from the adiabatic framework is determined by the way the nonadiabatic coupling matrix τ breaks up into blocks. In this article, we prove for the first time that the size of the diabatic matrix as obtained in a direct way is determined in the same way. In other words, if the dimension of the above-mentioned decoupled block is N, then the dimension of any diabatic potential energy matrix with physical relevance has to be N as well, regardless of how it was derived. This number, N, is also equal to the number of coupled diabatic−Schrödinger equations to be solved. (b) The second issue is, consequently, related to the actual required number of coupled Schrödinger equations to be solved to obtain a well-converged solution. Starting with the earlier introduced number N, we show that this number can be reduced, and in fact, it is most likely equal to the number of energetically open adiabatic states (for a given energy). While doing that, we rigorously derived the relevant diabatic potential matrix for this reduced case. We also worked out in detail an example related to a three-state case and derived the relevant 2 × 2 diabatic potential matrix.
Article
Reaction paths are studied with the help of diabatic potential-energy surfaces coupled in a generic external field. We show that all putative geometrical and topological features of two-dimensional (2D) potential-energy surfaces for an isomerization can be generated with a model consisting of strictly diabatic harmonic 2D wells coupled in a static (yet uniform) external field. For a large class of three-state models, we provide a phase diagram of possible topologies in the relevant parameter space for three-state models. By following the shift in diabatic electronic amplitudes (denoted by {|ck|2}) in coherent quantum states produced in the field, we assess whether the models within each phase can produce reaction paths visiting regions of configurational space that are structurally similar to the diabatic transition state.
Article
We present a new scheme for diabatizing electronic potential energy surfaces for use within the recently implemented direct-dynamics grid-based class of computational nuclear quantum dynamics methods, called Procrustes diabatization. Calculations on the well-studied molecular systems LiF and the butatriene cation, using both Procrustes diabatization and the previously implemented propagation and projection diabatization schemes, have allowed detailed comparisons to be made, which indicate that the new method combines the best features of the older approaches; it generates smooth surfaces, which cross at the correct molecular geometries, reproduces interstate couplings accurately, and hence allows the correct modeling of non-adiabatic dynamics.
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We intend to study the non-adiabatic interactions among the three lowest adiabatic states (1²A', 1²A'', and 2²A') of F+H2 triatomic reactive system in hyperspherical coordinates for a fixed hyperradius at ρ = 7.5 bohr as functions of hyperangles, θ (0° ≤ θ ≤ 90°) and (0° ≤ ≤ 360°). The adiabatic potential energy surfaces are calculated using MRCI level of methodology whereas the non-adiabatic coupling terms between those states are calculated from the analytic gradient methods implemented in MOLPRO quantum chemistry package. The ground (1²A') and the first excited (1²A'') states exhibit conical intersection (CI) and seam of CI along C2v geometries, whereas the first (1²A'') and the second (2²A') excited states undergo Renner-Teller coupling at linear geometries. We carry out adiabatic-to-diabatic transformation (ADT) by solving ADT equations to obtain ADT angles for constructing single-valued, continuous and symmetric 3 × 3 diabatic potential energy matrix so that subsequent accurate scattering calculations can be performed.
Article
We present a method for performing non-adiabatic, grid-based nuclear quantum dynamics calculations using diabatic potential energy surfaces (PESs) generated “on-the-fly”. Gaussian process regression is used to interpolate PESs by using electronic structure energies, calculated at points in configuration space determined by the nuclear dynamics, and diabatising the results using the propagation diabatisation method reported recently [J. Phys. Chem. A, 119, 12457 - 12470 (2015)]. Our new method is successfully demonstrated using a grid-based method to model the non-adiabatic dynamics of the butatriene cation. Overall, our scheme offers a route towards accurate quantum dynamics on diabatic PESs learnt on-the-fly.
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We calculate the adiabatic potential energy surfaces and non-adiabatic interactions among the three lowest singlet states (1¹A', 2¹A' and 3¹A') of H3⁺ in hyperspherical coordinates for a fixed hyperradius, ρ = 9 bohr as functions of hyperangles, θ (0 < θ < 90°) and (0 < < 360°). All ab initio calculations are performed using MRCI level of methodology implemented in quantum chemistry package, MOLPRO. The ground (1¹A') and the first excited (2¹A') states exhibit several conical intersections as functions of for θ > 70°. Subsequently, we carry out adiabatic to diabatic transformation (ADT) to obtain ADT angles for constructing single-valued, continuous, smooth and symmetric 3 × 3 diabatic potential energy matrix to perform accurate scattering calculations.
Article
While a wealth of studies exist on the potential energy surface (PES) for the F + H2 reaction, exploration of the effect of conical intersections (CIs on the corresponding adiabatic PES has only been initiated. In this respect, a recent emphasis on the specific role of the Jahn–Teller (JT) CI between the two lowest states has shown that the adiabatic-to-diabatic transformation (ADT) angle (mixing angle), γ12, between these two states deviates largely from its quantized value of π for configuration spaces with F close to the H–H system. Hence, investigating the effect of the other CIs present, involving the higher states, on the reactive channel is necessary. In this article, we report for the first time the collinear (2, 3), (3, 4) and (4, 5) JT CIs in this system and also reveal their major effects on γ12, which is instrumental in the diabatization of the PES for the title reaction system.
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The general features of the nonadiabatic coupling and its relation to molecular properties are surveyed. Some consequences of the ‘equation of motion’, formally expressing a ‘smoothness’ of a given molecular property within the diabatic basis, are demonstrated. A particular emphasis is made on the relation between a ‘smoothness’ of the electronic dipole moment and the generalized Mulliken–Hush formula for the diabatic electronic coupling.
Article
The role of conical intersections in the internal conversion (S1 → S0) of photoexcited ketene (H2CCO) is analyzed. The energy-minimized projection of a portion of the S1(1A‘ ‘)−S0(1A‘) seam of the conical intersection near the minimum energy-crossing point is studied as a function of the key internal coordinates R(C−C) and CCO. The characteristic parameters of the conical intersection points are used, to identify the two modes that evince the conical nature of the intersection, to determine the energy and singular part of the derivative coupling near the conical intersection, and to construct a transformation to diabatic states that rigorously removes the singularity in the derivative coupling. From the Franck−Condon region of the S0 → S1 excitation, barrierless paths were identified on S1 leading to Re(Ã1A‘ ‘), the equilibrium geometry of S1 ketene, and to Rmex, the minimum energy point on the S1−S0 seam of conical intersection. Following internal conversion onto S0 near Rmex, the barrierless paths leading to Re(X̃1A1), the equilibrium geometry of ground-state ketene, were found.
Article
The locus of the 11A′-21A′ seam of conical intersection for both HOH and OHH nuclear configurations in water is determined. Analytical expressions for the energies and derivative couplings in the vicinity of the conical intersections are presented. At each point of conical intersection a transformation to approximate diabatic states, based only on information about the derivative couplings, is defined. This transformation removes the preponderance of the derivative coupling near the conical intersection.
Article
We present the Molecular Symmetry (MS) adapted treatment of Non - Adiabatic Coupling Terms (NACTs) for the excited electronic states (2(2)E´ and 1(2) A´1 ) of Na3 cluster, where the adiabatic Potential Energy Surfaces (PESs) and the NACTs are calculated at the MRCI level by using ab initio quantum chemistry package (MOLPRO). The signs of the NACTs at each point of the Configuration Space (CS) are determined by employing appropriate Irreducible Representations (IREPs) arising due to MS group and such terms are incorporated into the Adiabatic to Diabatic Transformation (ADT) equations to obtain the ADT angles. Since those sign corrected NACTs and the corresponding ADT angles demonstrate the validity of Curl Condition for the existence of three - state (2(2)E´ and 1(2) A´1 ) sub - Hilbert space, it becomes possible to construct the continuous, single - valued, symmetric and smooth 3 × 3 diabatic Hamiltonian matrix. Finally, nuclear dynamics has been carried out on such diabatic surfaces to explore whether our MS based treatment of diabatization can reproduce the pattern of experimental spectrum for system B of Na3 cluster.
Article
We present a new approach to first-principles molecular dynamics that combines a general and flexible interpolation method with ab initio evaluation of the potential energy surface. This hybrid approach extends significantly the domain of applicability of ab initio molecular dynamics. Use of interpolation significantly reduces the computational effort associated with the dynamics over most of the time scale of interest, while regions where potential energy surfaces are difficult to interpolate, for example near conical intersections, are treated by direct solution of the electronic Schrödinger equation during the dynamics. We demonstrate the concept through application to the nonadiabatic dynamics of collisional electronic quenching of Li(2p). Full configuration interaction is used to describe the wave functions of the ground and excited electronic states. The hybrid approach agrees well with full ab initio multiple spawning dynamics, while being more than an order of magnitude faster. © 1999 American Institute of Physics.
Article
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In this article we present a survey of the various conical intersections which govern potential transitions between the three lower electronic states for the title molecular system. It was revealed that these three states, for a given fixed HH distance, RHH, usually form four conical intersections: two, between the two lower states and two, between the two upper states. One of the four is the well known equilateral D3h ci and the others are, essentially, C2v cis: One of them is located on the symmetry line perpendicular to the HH axis (just like the D3h ci) and the other two are located on both sides of this symmetry line and in this way form the ci-twins. The study was carried out for four RHH-values, namely, RHH = 0.74, 0.5417, 0.52, and 0.4777 Å. It was also established that there exists one single RHH-value designated as HH, located in the interval {0.52, 0.53 Å}, for which all four cis coalesce to become one kind of “super” ci which couples the three states. The numerical study was carried out employing the line integral approach for groups of two and three states. As for the two-state calculations we found that all D3h-cis, at close proximity, are circular (ordinary) Jahn-Teller-type cis, whereas all C2v-cis, at close proximity, are elliptic Jahn-Teller cis [Chem. Phys. Lett 354, 243 (2002)]. Particular attention is given to the 3-state quantization of the nonadiabatic coupling matrix. The quantization is found to be fulfilled in all situations as long as the regions in configuration space are not too far from the relevant cis. In the Discussion and Conclusion we discuss, among other subjects, the possibility to diabatize the adiabatic potential matrix. © 2003 American Institute of Physics.
Article
We report the first determination of a “most” diabatic basis for a triatomic molecule based exclusively on ab initio derivative couplings that takes careful account of the limitations imposed by the nonremovable part of those couplings. Baer [Chem. Phys. Lett. 35, 112 (1975)] showed that an orthogonal transformation from adiabatic states to diabatic states cannot remove all the derivative coupling unless the curl of the derivative coupling vanishes. Subsequently, Mead and Truhlar [J. Chem. Phys. 77, 6090 (1982)] observed that this curl does not, in general, vanish so that some of the derivative coupling is nonremovable. This observation and the historical lack of efficient algorithms for the evaluation of the derivative coupling led to a variety of methods for determining approximate diabatic bases that avoid computation of the derivative couplings. These methods neglect an indeterminate portion of the derivative coupling. Mead and Truhlar also observed that near an avoided crossing of two states the rotation angle to a most diabatic basis, i.e., the basis in which the removable part of the derivative coupling has been transformed away, could be obtained from the solution of a Poisson’s equation requiring only knowledge of the derivative couplings. Here a generalization of this result to the case of a conical intersection is used to determine a most diabatic basis for a section of the 1 1A′ and 2 1A′ potential energy surfaces of HeH2 that includes the minimum energy point on the seam of conical intersection. © 1998 American Institute of Physics.
Article
Recently there has been considerable interest, not to mention controversy, concerning a key aspect of the molecular Aharonov–Bohm (MAB) effect: the construction of the phase angle, induced by geometric phase effect, whose gradient is the vector potential characteristic of MAB theory. In the past this angle was constructed from explicit knowledge of the locus of the seam of conical intersection. Here it is shown how a phase angle that satisfies the requirements of MAB theory can be determined without a priori knowledge of the locus of points of conical intersection. This approach has important implications for direct dynamics. It is a corollary of a recent analysis that showed that diagonalizing the matrix of virtually any symmetric (real-valued Hermitian) electronic property operator in the subspace of states that intersect conically generates a transformation that removes all of the singularity of the derivative coupling at a conical intersection. Key aspects of this method are illustrated by considering the dipole moment operator near a point on the 1 3A″–2 3A″ seam of conical intersection in CH2. © 1999 American Institute of Physics.
Article
The local topography of a conical intersection can be represented by four parameters, readily determined from multireference configuration interaction wave functions, describing the pitch and tilt of the double cone. The time-dependent Schrödinger equation is solved in the vicinity of a conical intersection in the adiabatic basis using an approach tailored to this representation. It is shown that an adiabatic state treatment, which offers conceptual advantages is, in the appropriate set of internal coordinates, not qualitatively more difficult than the equivalent calculation in a diabatic basis. The present treatment is fully hermitian and takes full account of the geometric phase effect being, for example, gauge invariant (in the infinite basis limit) and could be used to develop a fully adiabatic description of nonadiabatic dynamics. The gauge invariant formulation provides interesting insights into the consequences of neglecting the geometric phase. The algorithm is used to study the effects of the double cone’s topography on the outcome of a nonadiabatic transition. Transitions from both the upper state to the lower state and from the lower state to upper state are considered for representative sets of conical parameters. The effects of the local topography on the outcome of nonadiabatic transitions can be dramatic. © 2001 American Institute of Physics.
Article
Portions of the S1[1A″(2 1A)]–S0[1A′(1 1A)] seam of conical intersection relevant to the internal conversion S1→S0 of photoexcited isocyanic acid HNCO are analyzed. The topography of the potential energy surfaces, and the derivative coupling, in the vicinity of a conical intersection is described in terms of four conical parameters. These parameters are also used to obtain a local diabatic representation that removes the singularity in the derivative coupling. Continuity is achieved through the use of a recently described orthogonalization procedure. The conical parameters demonstrate that the double cones of concern are significantly tilted, which has important implications for the nuclear dynamics. © 2001 American Institute of Physics.
Article
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A method for constructing diabatic potential energy matrices by interpolation of ab initio quantum chemistry data is described and tested. This approach is applicable to any number of interacting electronic states, and relies on a formalism and a computational procedure that are more general than those presented previously for the case of two electronic states. The method is tested against an analytic model for three interacting electronic states of NH(3) (+).
Article
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This work considers the adiabatic-diabatic transformation for electronic states and the diabatic representation which follows accordingly. Two cases where ambiguity is encountered are discussed: one is the case where the reduced electronic manifold is not well separated everywhere in configuration space and the other is the case of conical intersections exemplified by the E ε Jahn-Teller situation. It is shown that in both cases well defined diabatic states can be formed in most regions of configuration space.
Article
In the Born-Oppenheimer approximation for molecular dynamics as generalized by Born and Huang, nuclei move on multiple potential-energy surfaces corresponding to different electronic states. These surfaces may intersect at a point in the nuclear coordinates with the topology of a double cone. These conical intersections have important consequences for the dynamics. When an adiabatic electronic wave function is transported around a closed loop in nuclear coordinate space that encloses a conical intersection point, it acquires an additional geometric, or Berry, phase. The Schr{umlt o}dinger equation for nuclear motion must be modified accordingly. A conical intersection also permits efficient nonadiabatic transitions between potential-energy surfaces. Most examples of the geometric phase in molecular dynamics have been in situations in which a molecular point-group symmetry required the electronic degeneracy and the consequent conical intersection. Similarly, it has been commonly assumed that the conical intersections facilitating nonadiabatic transitions were largely symmetry driven. However, conical intersections also occur in the absence of any symmetry considerations. This review discusses computational tools for finding and characterizing the conical intersections in such systems. Because these purely accidental intersections are difficult to anticipate, they may occur more frequently than previously thought and in unexpected situations, making the geometric phase effect and the occurrence of efficient nonadiabatic transitions more commonplace phenomena. {copyright} {ital 1996 The American Physical Society.}
Article
We derive the most general form of the first terms of the power-series expansion of the electronic Hamiltonian in the neighborhood of the conical intersection at the equilateral triangle configuration of the homonuclear triatomic system M3. Previous treatments of this problem had assumed that the derivative coupling between Born–Oppenheimer states could be transformed away by choosing a strictly diabatic basis of electronic states. It has recently been pointed out, however, that this is not possible, in general. Making full use of the symmetry of the problem, and also taking account of the molecular Aharonov–Bohm effect, we obtain explicitly the leading terms of the expansion for electronic energies and wave functions, and of the derivative coupling. In terms of the expansion parameter r, a measure of the distance from the equilateral triangle configuration, the derivative coupling can be transformed away through the first order, but in the second order, nonremovable terms appear which are expected to be important in some problems.
Article
The 1, 2 1A′ potential energy surfaces (PESs) of the He–H2 system, surfaces which correlate asymptotically with He(1S)+H2(X 1Σ+g, B 1Σ+u) system states, are characterized using MCSCF/CI wave functions. The existence of charge transfer structures of the form (HeH)+–H− on the two PESs is considered as are the electronic structure aspects of the nonadiabatic quenching process He+H2(B 1Σ+u )→He+H2(X 1Σ+g). While this work builds on previously reported theoretical treatments of these PESs, both qualitative and quantitative differences are found. In particular, our predicted entrance channel saddle point corresponds to a barrier of 1.5 kcal/mol on the 2 1A′ PES which is significantly lower than previous work. More significantly an extended region of large nonadiabatic effects characterized by the near degeneracy of the 1 1A′ and 2 1A′ PESs, E(2 1A′)−E(1 1A′)<0.5 kcal/mol, has been located. This region of the 2 1A′ PES, which is exothermic with respect to dissociation to He+H2(B 1Σ+u) and is characterized by general Cs, rather than C2v or C∞v geometries, was not uncovered in previous studies. Analyses based on the molecular dipole moment and the nonadiabatic coupling matrix elements 〈Ψ(2 1A′)‖(∂/∂Rα) Ψ(1 1A′)〉 are used to characterize this region.
Article
A recently developed method for determining avoided surface crossings using analytic gradient techniques is used to locate an actual crossing seam for the 1 1A′ and 2 1A′ potential energy surfaces of the He–H2 system. This seam is not related to any high symmetry nuclear configurations. The computational procedure, which is based on the minimization of ΔEIJ(R)2≡[EI(R)−EJ(R)]2, &(R)]2, uses different density matrices to simplify the construction of the energy difference gradient, the most costly step in the procedure. The actual crossing seam, R(r), is specified by the ordered triple R(r)≡[R(r), γ(r),r] for which ΔEIJ(R)=0. It is exoergic with respect to the He–H2(B 1Σ+u) asymptote for r≊[2.60, 5.70]. Here r≡R(H2), R=R(He–H2) and γ is the He–H2 angle. This seam defines a region nuclear coordinate space near which helium can efficiently quench H2(B 1Σ+u).
Article
Conical intersections complicate the computational treatment of nuclear dynamics in the adiabatic state basis through the geometric phase effect and singularities in the derivative couplings. The diabatic representation seeks to eliminate these difficulties. However, the adiabatic to diabatic state transformation is necessarily approximate in a polyatomic molecule since the derivative couplings cannot be rigorously removed. This point is rarely considered when constructing approximate diabatic states. The nonremovable part of the derivative couplings is investigated by considering the integral of the derivative coupling along closed loops in the vicinity of the 1 2A′–2 2A′ seam of conical intersections in H3. © 1996 American Institute of Physics.
Article
In this work some aspects of the atom-molecule interactions are extended to include electronic transitions. The main emphasis is directed towards the close relationship between the adiabatic and the diabatic representations. We show how one may transform from the adiabatic scheme to the diabatic one without losing physical information and with minimal amount of numerical efforts. The case of two surfaces (or two electronic states) is treated in particular detail. The main outcome of this study is that, although the electronic information regarding the atom-molecule interaction is given in the adiabatic scheme, one should transform to the diabatic scheme when treating the nuclear interactions.