Five Electronic State Beyond BornâOppenheimer Equations and Their

Article pubs.acs.org/JPCA

Five Electronic State Beyond Born−Oppenheimer Equations and Their Applications to Nitrate and Benzene Radical Cation Soumya Mukherjee, Bijit Mukherjee, and Satrajit Adhikari* Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India S Supporting Information *

ABSTRACT: We present explicit form of Adiabatic to Diabatic Transformation (ADT) equations and expressions of non-adiabatic coupling terms (NACTs) for a coupled five-state electronic manifold in terms of ADT angles between electronic wave functions. ADT matrices eliminate the numerical instability arising from singularity of NACTs and transform the adiabatic Schrödinger equation to its diabatic form. Two real molecular systems NO3 and C6H+6 (Bz+) are selectively chosen for the demonstration of workability of those equations. We examine the NACTs among the lowest five electronic states of the NO3 radical [X̃ 2A2′ (12B2), Ã 2E″ (12A2 and 12B1) and B̃ 2E′ (12A1 and 22B2)], in which all types of non-adiabatic interactions, that is, Jahn−Teller (JT) interactions, Pseudo Jahn−Teller (PJT) interactions, and accidental conical intersections (CIs) are present. On the other hand, lowest five electronic states of Bz+ [X̃ 2E1g (12B3g and 12B2g), B̃ 2E2g (12Ag and 12B1g), and C̃ 2A2u (12B1u)] depict similar kind of complex feature of non-adiabatic effects. For NO3 radical, the two components of degenerate in-plane asymmetric stretching mode are taken as a plane of nuclear configuration space (CS), whereas in case of Bz+, two pairs are chosen: One is the pair of components of degenerate in-plane asymmetric stretching mode, and the other one is constituted with one of the components each from out-of-plane degenerate bend and in-plane degenerate asymmetric stretching modes. We calculate ab initio adiabatic potential energy surfaces (PESs) and NACTs among the lowest five electronic states at the CASSCF level using MOLPRO quantum chemistry package. Subsequently, the ADT is performed using those newly developed equations to validate the positions of the CIs, evaluate the ADT angles and construct smooth, symmetric, and continuous diabatic PESs for both the molecular systems. (adiabatic) acquire a phase and invert their sign.18 This multivaluedness property of the eigenfunctions was tested using Jahn−Teller (JT) CI model by Herzberg and LH and was corrected in an ad hoc manner.19 Later on, Mead and Truhlar20 removed the multivaluedness of the electronic wave functions and generalized BO equations by introducing a vector potential into the nuclear Hamiltonian. Varandas, Tennyson, and Murrell21 applied the LH theorem on a realistic triatomic system LiNaK in which it was shown that the dominant coefficient of the electronic eigenfunctions undergoes a sign change along a closed contour around a CI. On the other hand, the Hellmann−Feynman theorem depicts the possibility of singularity in NACTs22,23 over the nuclear CS whenever electronic states become degenerate or near degenerate. Thus, it becomes necessity to remove any such singularity(s) from the equations, which appear in the kinetic energy operator in the SE, by transforming to a different representation or basis, known as the diabatic one. In this new representation, those coupling terms appear as the off-diagonal terms in the diabatic potential energy matrix and are also smoothly varying functions of nuclear coordinates. Such transformation is a unitary one and

1. INTRODUCTION The solution of the Schrödinger Equation (SE) provides various experimental observable quantities like reactive/nonreactive cross sections, spectroscopic quantities, etc. While solving the molecular SE, the Born−Oppenheimer (BO) approximation1,2 stands out to be an excellent starting point that separates the mechanics of fast-moving electrons from the slow-moving nuclei on the basis of huge differences in masses. On the contrary, the approximation does not hold good in many situations whenever excited electronic states are involved in a given molecular process. Processes involving charge-transfer reactions, measurements of scattering cross sections, and photochemical reactions show profound discrepancies between calculated single and multiple surface results with experimental observations.3−13 In such situations, the non-adiabatic coupling terms (NACTs) become quite significant and may not be overlooked14−17 from the SE as being done while implementing BO approximation. Though adiabatic potential energy surfaces (PESs) and NACTs are the two major outcomes from the BO treatment, yet there are few major issues, which are rather fundamental in nature, need to be taken care from first principle. In the 1960s, Longuet-Higgins (LH) observed while surrounding a point of conical intersection (CI) adiabatically in the nuclear configuration space (CS), the BO eigenfunctions © XXXX American Chemical Society

Received: May 12, 2017 Revised: July 22, 2017 Published: July 24, 2017 A

DOI: 10.1021/acs.jpca.7b04592 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

the workability of our developed equations. However, Vértesi et al. have formulated ADT equation of one privileged angle (γ12) for sub-Hilbert spaces up to five electronic states and explored on H + H2 system.43 Before discussing the details of the BO treatment for five-state sub-Hilbert space, we present the brief account of the non-adiabatic interactions prevailing in those molecular systems. Diverse complexities in electronic structure make nitrate radical (NO3) as one of the most intricate molecules in the perspective of non-adiabatic chemistry. The five electronic states of this radical, namely, X̃ 2A′2 (12B2), Ã 2E″ (12A2 and 12B1), and B̃ 2E′ (12A1 and 22B2), are within an energy range of 2 eV at the equilibrium geometry and are heavily vibronically coupled to each other in the Franck−Condon region of nuclear CS. From the early 1980s, several spectroscopic methods like dispersed fluorescence,44,45 Fourier-transform infrared absorption,46,47 photoelectron spectroscopy,48 and cavity ringdown spectroscopy (CRDS)49,50 have been implemented to explore the characteristics of the complex spectral features exhibited by NO3. Weaver et al.48 failed to assign some of the features associated with the photoelectronic spectra, but CRDS experiments49 indicated a strong evidence in favor of JT coupling effect operating within Ã 2E″ state, which demanded necessary theoretical authentication. Though the earlier quantum mechanical studies51−54 were unable to predict the actual symmetry of the equilibrium geometry unambiguously, Eisfeld and Morokuma55 finally established it to be of D3h symmetry. A vibronic coupling model was introduced by Mayer et al.56 to explain the observed spectroscopic profile, and this was modified subsequently by the same group57 incorporating linear and bilinear coupling terms and then by Faraji et al.58 including thirdand fourth-order coupling elements in the modified Hamiltonian. Stanton and co-workers59,60 carried out detailed study on the vibronic structure of this radical and could able to unravel the salient features of the observed spectra using the predicted coupling between the electronic states. The theoretical calculations so far show that Ã 2E″ and B̃ 2E′ are subjected to JT interactions at the equilibrium D3h point.61 In addition, the ground state X̃ 2A′2 undergoes strong pseudo Jahn−Teller (PJT) distortion due to the B̃ 2E′ state.61 These interactions are expected to strongly affect the dynamics of the radical and the associated features of the photoelectron spectra of NO−3 anion.48 The photophysics of Bz+ radical cation drew attention over the past few decades owing to its complex electronic structure as a consequence of dramatic failure of Born−Oppenheimer approximation for this system. The diffuse appearance of B̃ 2E2g-C̃ 2A2u band in photoelectron spectra was first illustrated by Köppel et al.62 and then subsequently, the X̃ 2E1g state was taken into account to depict the combined multimode JT splitting.63 Later on several theoretical studies pointed out good agreement between the ab initio calculated photoelectronic spectra64 and experimentally obtained diffused D̃ 2E1u-Ẽ 2B2u band. In all theoretical treatments62−70 the electronic states, namely, X̃ 2E1g, B̃ 2E2g, C̃ 2A2u, D̃ 2E1u, and Ẽ 2B2u along with the CIs among the PESs are investigated for its dynamical studies. Goode et al.71 demonstrated complex vibrational structure of B̃ 2E2g state using combination of Herzberg−Teller and PJT coupling between this state and close-lying C̃ 2A2u state. JT coupling parameters for e2g vibrational modes of this radical cation have been calculated by Johnson,72 and those are used to illustrate the photoinduced Rydberg ionization spectra of B̃ 2E2g state.73 The low-lying doublet states of Bz+, B̃ 2E2g, and C̃ 2A2u exhibit intense non-adiabatic interactions62 among them. The JT effects within

is known as the Adiabatic-to-Diabatic transformation (ADT). The NACT subtends vector field that can be decomposed into longitudinal and transverse components,24 where the former is expressed as the derivative of a scalar, and the latter by the curl of a vector. By performing ADT, the longitudinal component can be eliminated,14,25 which is known as the removable part of the NACT. On the other hand, the transverse or the nonremovable part of the couplings may be neglected if it becomes insignificant at the close proximity of a CI or can be minimized by including sufficient number of electronic states to form a sub-Hilbert space over the range of interested nuclear configuration space. In this context, Baer et al.26 have proposed a method of reducing diabatic potential matrix as per the required dimension of sub-Hilbert space and shown its formulation for 2 × 2 diabatic potential matrix from a coupled threestate system. Hobey and McLachlan27 were among the first who used the technique of ADT for a single degree of freedom to eliminate NACTs from the BO close-coupled equations. This was followed by F. T. Smith28 who carried out the same technique for a diatomic molecule. On the other hand, M. Baer derived a general formulation regarding the diabatization of two adiabatic PESs for processes involving triatomic collisions. In that work, Baer devised the ADT condition that led to the determination of the ADT matrix elements by solving the differential equations for ADT angles along a two-dimensional contour.14,25 The existence and the uniqueness of the solution to those equations were predicted to be guaranteed by the validity of the so-called “curl condition”.25,29 In general, the ADT angles can be evaluated by integrating the NACTs around the CI point(s), where the closed contour integrals of those couplings will give magnitudes in multiples of π (pi).30,31 The satisfaction of curl conditions by the NACTs also validates the existence of the sub-Hilbert space, that is, an isolated group of states forming a complete space. Though the above said condition is not valid at the CI point(s), nevertheless, the NACTs can be removed from the SE if the line integrals are quantized forming a subspace. In other words, this quantization rule is obeyed only when all electronic states constituting the sub-Hilbert space are taken into account.32 Adhikari et al. have generalized the BO treatment for three as well as four coupled electronic states in terms of electronic basis functions or the ADT matrix elements, where the explicit forms of the NACTs, their Curl-Divergence equations, Curl conditions, and the diabatic PESs in terms of the ADT angles have been formulated.33−35 Not only this approach paves a practical way of handling NACTs with singularities at the CI point(s) in the nuclear CS but also has been quite successfully implemented on model33−36 as well as realistic37−40 systems in constructing smooth, continuous, and single-valued diabatic PESs. Later on, ADT equations for four-state sub-Hilbert space have been successfully implemented to investigate the non-adiabatic interactions in HCNH molecule.41,42 In this regard, it is worthwhile to mention that till now no extensive formalism of ADT for five-state sub-Hilbert space has been developed. Nevertheless, molecular systems having strong vibronic interaction between five electronic states do exist. NO3 radical and C6H+6 (Bz+) radical cation are two examples of such systems. In our work, we will pursue the formulation of beyond BO equations for five-state sub-Hilbert space and present for the first time the explicit form of ADT angles in terms of NACTs and NACTs in terms of the ADT angles for constructing five-sheeted diabatic PESs. In the cases of NO3 radical and Bz+ radical cation, we test B


Article

The Journal of Physical Chemistry A the ground, X̃ 2E1g as well as the excited state, B̃ 2E2g are quite predominant particularly making the ground-state equilibrium structure of Bz+ to lower symmetry D2h rather than D6h.65,72 Besides the JT effects, there are accidental CIs and PJT interactions occurring between B̃ 2E2g and C̃ 2A2u states.62 In this work, we investigate the JT interactions of the ground (X̃ 2E1g) and excited state (B̃ 2E2g) as well as accidental CIs within the higher excited states (B̃ 2E2g-C̃ 2A2u) using the newly developed ADT equations within the Beyond BO (BBO) formalism. The present article is organized as follows: Section 2 gives the detailed theoretical development of BBO equations for fivestate sub-Hilbert space. In that section, we will present the explicit form of the ADT equations along with the NACTs as expressed in terms of the ADT angles. In Section 3, we will discuss the ab initio calculations performed for the two molecular systems NO3 and Bz+, while in Section 4 the obtained results will be thoroughly analyzed. Finally, we lay out the summary of the present work in Section 5.

where Uij = uiδij and τ⃗ is the non-adiabatic coupling matrix (NACM) defined as τij⃗ = ⟨ξi(se|sn)|∇⃗ξj(se|sn)⟩

The NACM, τ⃗ is a skew-symmetric matrix if the electronic eigenfunctions are chosen to be real. In the case for a five-state sub-Hilbert space, the form of the NACM will be ⎛ 0 τ12⃗ τ13⃗ τ14⃗ ⎜ ⎜−τ12⃗ 0 τ23 τ24 ⃗ ⃗ ⎜ 0 τ34 τ ⃗ = ⎜ −τ13⃗ −τ23 ⃗ ⃗ ⎜ 0 ⃗ −τ34 ⃗ ⎜−τ14⃗ −τ24 ⎜ ⃗ −τ35 ⃗ − τ45⃗ ⎝ −τ15⃗ −τ25

5

∑ ψi(sn)|ξi(se|sn)⟩ i=1

(1)

ψ = Aϕd

The total wave function satisfies the time-independent Schrödinger equation (SE): ̂ se , sn)|Ψ(se , sn)⟩ = E|Ψ(se , sn)⟩ H(

(2)

(3)

W = A†UA

(4)

(12)

known as the adiabatic-to-diabatic transformation (ADT) condition.14 The Lie algebra of skew-symmetric matrix τ⃗ (5 × 5 NACM) belongs to SO(5) nonabelian group and predicts that the ADT equation will give rise to solutions that will be path-dependent due to choice of integration contour over the nuclear coordinates. Thus, the ADT matrices obtained by various paths of integration will be in general different from each other, but they will be related to one another by orthogonal transformations. To illustrate the situation, let us take two ADT matrices A1 and A2, obtained from solutions due to two different choices of contour. It can be shown that the product of these two matrices can be written as37

(5)

where ui(sn) are the adiabatic PESs. When the form of total wave function defined in eq 1 is substituted in the molecular SE [eq 2] followed by projections with various electronic eigenfunctions, we obtain a set of kinetically coupled nuclear SEs. In matrix notation this can be represented as ⎯ ⎤ ⎡ 1 → 2 ⎢⎣ − 2 ( ∇n + τ ⃗) + U − E⎥⎦ψ = 0

(11)

under the condition → ⎯ ∇nA + τ ⃗ A = 0

The electronic wave functions (|ξi(se|sn)⟩) are the eigenfunctions of Ĥ e(se|sn) operator. They are parametrically dependent on nuclear coordinates, and as a result, the eigenvalues of electronic Hamiltonian are functions of nuclear coordinates. The electronic eigenfunctions |ξi(se|sn)⟩ that form the basis of the BO expansion are the solutions of the electronic Schrödinger equation: Ĥ e(se|sn)|ξi(se|sn)⟩ = ui(sn)|ξi(se|sn)⟩

(9)

where ψ and ϕ represent the nuclear wave functions in adiabatic and diabatic representations, respectively. With such a modification, the nuclear SE in diabatic representation takes the following form: 1 − ∇2nϕd + (W − E)ϕd = 0 (10) 2 where

with T̂ n is the nuclear kinetic energy operator and Ĥ e(se|sn) is the electronic Hamiltonian. The nuclear kinetic energy operator takes the form ⎛ ∇2 ⎞ 1 s ,i ̂ Tn = − ∑ ⎜⎜ n ⎟⎟ 2 i ⎝ mi ⎠

(8)

d

where the molecular Hamiltonian is given by Ĥ (se , sn) = T̂n(sn) + Ĥ e(se|sn)

τ15⃗ ⎞ ⎟ τ25 ⃗ ⎟ ⎟ τ35 ⃗ ⎟ ⎟ τ45⃗ ⎟ ⎟ 0⎠

While performing numerical calculations, since it is difficult to work with NACTs due to their singular nature at the point of degeneracies, that is, CI points, we straightaway come across the demerits of the adiabatic representation. This motivates us to search for another representation in which the coupling terms among the electronic states appear in the form of potential commonly known as diabatic coupling. In other words, any singularity in the kinetic coupling can be hidden in the diabatic PESs through ADT angles. Going to the diabatic representation requires an orthogonal rotation from the adiabatic one symbolized by the adiabatic-to-diabatic transformation matrix, A. The nuclear wave functions are then related to each other by

2. THEORY In this section we carry out first principle based BO treatment for any five-state electronic sub-Hilbert space assuming the presence of CIs anywhere in the given nuclear configuration space. Since it is assumed that the five states in consideration are decoupled from the rest of the excited states of a molecular system, the total wave function can be expanded in terms of a finite electronic basis set in the spirit of the BO expansion:1,2 |Ψ(se , sn)⟩ =

(7)

A1†A 2 = A†(p0 , q0)exp[−ΔpΔq(Curl τ )pq ]A(p0 , q0)

(13)

where p0 and q0 are the initial nuclear coordinates and Δp and Δq are small increments on the contour. Thus, the two matrices will differ from each other whenever the Curl τ is nonzero

(6) C


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The Journal of Physical Chemistry A (zero for two-state sub-Hilbert space). On the other hand, the product matrix (B = A†1 A2) is orthogonal as shown below:37

∇⃗θ14 = − τ45⃗ cos θ14 tan θ15 − cos θ12 cos θ13 × (τ14⃗ + τ15⃗ sin θ14 tan θ15) + sin θ12 cos θ13(τ24 ⃗

B†B = A†(p0 , q0)exp[ΔpΔq(Curl τ )pq ]

+ τ25 ⃗ sin θ14 tan θ15) + sin θ13(τ34 ⃗ + τ35 ⃗ sin θ14 tan θ15) (16c)

†

× A(p0 , q0)A (p0 , q0)exp[−ΔpΔq(Curl τ )pq ] × A(p0 , q0) = I

∇⃗θ15 = − τ15⃗ cos θ12 cos θ13 cos θ14 + τ25 ⃗ sin θ12 cos θ13 cos θ14

(14)

+ τ35 ⃗ sin θ13 cos θ14 + τ45⃗ sin θ14

The above equation implies that A1 and A2 are related through an orthogonal transformation matrix B, and hence the corresponding diabatic Hamiltonian will also be related through such transformation (B) ensuring the calculated observable to be path-independent. The model form of A can be chosen keeping in mind that it must be orthogonal and that the elements are cyclic with respect to a parameter. For five-state sub-Hilbert space, in order to fulfill the orthonormality condition, 15 independent relations are to be satisfied, and therefore we need 10 rotation angles, which are the adiabatic-to-diabatic transformation angles. The A matrix can be constructed by multiplying 10 rotation matrices15 in the following way:

τ15⃗ = −∇⃗θ15 cos θ12 cos θ13 cos θ14 + ∇⃗θ25 cos θ15 cos θ24{−sin θ12 cos θ23 + cos θ12(sin θ13 sin θ23 + cos θ13 sin θ14 tan θ24)} − ∇⃗θ35 cos θ15 cos θ25[sin θ12(sin θ23 cos θ34 − cos θ23 sin θ24 sin θ34) + cos θ12 × {sin θ13 cos θ23 cos θ34 + ( −cos θ13 sin θ14 cos θ24 + sin θ13 sin θ23 sin θ24)sin θ34}]

A = A12(θ12) ·A13(θ13) ·A14(θ14) ·A15(θ15) ·A 23(θ23) · A 24(θ24) ·A 25(θ25) ·A34(θ34) ·A35(θ35) ·A45(θ45)

− ∇⃗θ45 cos θ15 cos θ25 cos θ35

(15)

× [sin θ12(cos θ23 sin θ24 cos θ34 + sin θ23 sin θ34)

This multiplication of the rotation matrices can be done in 10! different ways, but any particular order will provide numerically same diabatic potential energy matrix.34 Any Aij matrix in the product at the right-hand side (R.H.S.) of the above equation depicts the coupling between ith and jth electronic states. For example, A12(θ12) matrix is ⎛ cos θ12 sin θ12 ⎜ ⎜−sin θ12 cos θ12 A12 = ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎝ 0 0

+ cos θ12{cos θ13 sin θ14 cos θ24 cos θ34 + sin θ13(− sin θ23 sin θ24 cos θ34 + cos θ23 sin θ34)}] (17a)

τ25 ⃗ = ∇⃗θ15 sin θ12 cos θ13 cos θ14

0 0 0⎞ ⎟ 0 0 0⎟ ⎟ 1 0 0⎟ 0 1 0⎟ ⎟ 0 0 1⎠

− ∇⃗θ25 cos θ15 cos θ24{cos θ12 cos θ23 + sin θ12(sin θ13 sin θ23 + cos θ13 sin θ14 tan θ24)} + ∇⃗θ35 cos θ15 cos θ25{−cos θ12 sin θ23 cos θ34 + sin θ12(− cos θ13 sin θ14 cos θ24

The general rule of writing such a form of the matrices in eq 15 is for Aij (θij), ii and jj elements become cos θij; ij and ji elements become sin θij and −sin θij, respectively, while the remaining diagonal and the off-diagonal terms will be one and zero, respectively. Substituting the model form of rotation matrix, A and the antisymmetric form of NACM in eq 12, some mathematical manipulation leads to the explicit ADT equations and expressions of τ for five-state sub-Hilbert space. Among them, some of the differential equations representing the mixing angles between ground state and the four excited states and NACTs (τ) corresponding to the couplings between fifth state and the first four electronic states are presented here:

+ sin θ13 sin θ23 sin θ24)sin θ34 + cos θ23(sin θ12 sin θ13 cos θ34 + cos θ12 sin θ24 sin θ34)} + ∇⃗θ45 cos θ15 cos θ25 cos θ35 × {sin θ12 cos θ13 sin θ14 cos θ24 cos θ34 + sin θ12 sin θ13( −sin θ23 sin θ24 cos θ34 + cos θ23 sin θ34) − cos θ12(cos θ23 sin θ24 cos θ34 + sin θ23 sin θ34)}

∇⃗θ12 = − τ12⃗ − sin θ12(τ13⃗ tan θ13 + τ14⃗ sec θ13 tan θ14

(17b)

τ35 ⃗ = ∇⃗θ15 sin θ13 cos θ14 + ∇⃗θ25 cos θ13 cos θ15 cos θ24

+ τ15⃗ sec θ13 sec θ14 tan θ15)

× (sin θ23 − tan θ13 sin θ14 tan θ24)

− cos θ12{τ23 ⃗ tan θ13 + sec θ13(τ24 ⃗ tan θ14 + τ25 ⃗ sec θ14 tan θ15)}

(16d)

− ∇⃗θ35 cos θ13 cos θ15 cos θ25{cos θ23 cos θ34 (16a)

+ sin θ34(sin θ23 sin θ24 + tan θ13 sin θ14 cos θ24)} + ∇⃗θ45 cos θ13 cos θ15 cos θ25 cos θ35{

∇⃗θ13 = − cos θ13(τ34 ⃗ tan θ14 + τ35 ⃗ sec θ14 tan θ15) + sin θ12(τ23 ⃗ + τ24 ⃗ sin θ13 tan θ14 + τ25 ⃗ sin θ13 sec θ14 tan θ15)

−cos θ23 sin θ34 + cos θ34(sin θ23 sin θ24

− cos θ12{τ13⃗ + sin θ13(τ14⃗ tan θ14 + τ15⃗ sec θ14 tan θ15)}

+tan θ13 sin θ14 cos θ24)}

(16b) D

(17c) DOI: 10.1021/acs.jpca.7b04592 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article


and Q19−Q16x, as the nuclear configuration spaces. The modes Q16x and Q16y of Bz+ represent the two components of the degenerate in-plane asymmetric stretching mode (ν16), and Q19 is one of the two components of the degenerate out-of-plane CH bend mode (ν19) of Bz+. It may be noted that these two modes (ν16,ν19) are quite important for depicting nonadiabaticities of the radical cation Bz+ as predicted by previous theoretical calculations.62 The experimental frequencies of these two modes (ν16 and ν19) for the neutral ground state (Bz) are 1609 and 967 cm−1, and these belong to e2g and e2u symmetries, respectively, in D6h point group. The naming of these modes is consistent with that done by Köppel et al.62 Both in Q16x−Q16y and Q19−Q16x configuration spaces, we have employed CASSCF methodology to calculate the adiabatic PESs for the lowest five doublet states of Bz+, specifically, X̃ 2E1g (12B3g and 12B2g), B̃ 2E2g (12Ag and 12B1g), and C̃ 2A2u (12B1u). The labels outside the parentheses denote the states in D6h symmetry, whereas inside the parentheses those are represented in D2h symmetry. The ab initio calculations are done with an active space of 29 electrons distributed over 15 orbitals (29e, 15o) with Gaussian basis set 6-31g**. For the calculation of NACTs between different pairs of states, we have employed CP-MCSCF methodology implemented in MOLPRO. The calculated NACTs reveal the prominent non-adiabatic interactions present within the nuclear configuration spaces in which we intend to perform ADT to get the diabatic PESs. All the ab initio findings and the ADT results are thoroughly discussed in Section 4.2.

τ45⃗ = ∇⃗θ15 sin θ14 + ∇⃗θ25 cos θ14 cos θ15 sin θ24 + ∇⃗θ35 cos θ14 cos θ15 cos θ24 cos θ25 sin θ34 − ∇⃗θ45 cos θ14 cos θ15 cos θ24 cos θ25 cos θ34 cos θ35 (17d)

All differential equations for ADT angles and expressions of τ are given in Supporting Information. We can express the differential equations for ADT angles in matrix form: ⎛ ∇⃗θ ⎞ ⎜ 12 ⎟ ⎜ ⃗ ⎟ ⎛ ⋯ ⎜ ∇θ13 ⎟ ⎜ Ω11 Ω12 Ω13 ⎜ ⃗ ⎟ ⎜Ω Ω 22 ⎜∇θ23 ⎟ ⎜ 21 ⎜ ⎟ ⎜Ω ⃗ ⎜ ∇θ14 ⎟ ⎜ 31 ⎜ ⎟ ⎜ ⋱ ⎜∇⃗θ24 ⎟ ⎜ ⋮ ⎜ ⎟=⎜ ⎜∇⃗θ34 ⎟ ⎜ ⎜ ⎟ ⎜ ⋮ ⎜ ∇⃗θ15 ⎟ ⎜ ⎜ ⎟ ⎜ ⎜∇⃗θ ⎟ ⎜ Ω 91 25 ⎜ ⎟ ⎜ ⎜ ∇⃗θ ⎟ ⎝ Ω10,1 Ω10,2 ⋯ 35 ⎜ ⎟ ⎜ ⃗ ⎟ ⎝ ∇θ45 ⎠

Ω19

⋱ Ω10,9

Ω1,10 ⎞ ⎟ Ω 2,10 ⎟ ⎟ ⎟ ⎟ ⎟ ⋮ ⎟. ⎟ ⎟ ⋮ ⎟ ⎟ ⎟ Ω 9,10 ⎟ ⎟ Ω10,10 ⎠

⎛ τ12⃗ ⎞ ⎜ ⎟ ⎜ τ13⃗ ⎟ ⎜ ⎟ ⃗ ⎟ ⎜ τ23 ⎜ ⎟ ⎜ τ14⃗ ⎟ ⎜ ⎟ ⃗ ⎟ ⎜ τ24 ⎜ ⎟ τ ⃗ ⎟ ⎜ 34 ⎜τ⃗ ⎟ ⎜ 15 ⎟ ⎜τ ⃗ ⎟ ⎜ 25 ⎟ ⎜ τ35 ⃗ ⎟ ⎜ ⎟⎟ ⎜ ⎝ τ45⃗ ⎠

(18)

Elements of Ω matrix are presented in Supporting Information. Elements of A matrices can be obtained at each and every nuclear configuration while integrating the ADT equations in terms of NACTs calculated by MOLPRO quantum chemistry package.74 When we perform similarity transformation with this numerically calculated A on [eq 11], we get smooth, singlevalued, continuous diabatic potential energy surface.

4. RESULTS AND DISCUSSION 4.1. NO3 Radical. The Q3x and Q3y coordinates represent the two components of degenerate in-plane asymmetric stretching mode (ν3) of NO3 radical. We transform these two coordinates into polar form to exploit the inherent C3 rotational symmetry present in the system as

3. AB INITIO CALCULATIONS FOR PESS AND NACTS 3.1. NO3 Radical. We vary two coordinates of degenerate components of the in-plane asymmetric stretching vibrational mode (ν3) of NO3 (symbolized as Q3x−Q3y plane) to perform the ab initio calculations of the lowest five doublet states of NO3, namely, X̃ 2A′2 (12B2), Ã 2E″ (12A2 and 12B1), and B̃ 2E′ (12A1 and 22B2). The labels outside the parentheses denote the states in D3h symmetry, whereas inside the parentheses they are represented in C2v symmetry. We implemented Complete Active Space Self-Consistent Field (CASSCF) level of calculations implemented in MOLPRO quantum chemistry package74 to calculate the adiabatic PESs. The ab initio data points are obtained in C1 symmetry using Gaussian basis set 6-31g**. The state-averaged CASSCF calculations have been done employing a nine electrons in eight orbitals (9e, 8o) configuration active space (CAS). The CAS comprising of 2352 configuration state functions reproduces the degeneracies at the D3h point of the molecule with (Cs) or without (C1) symmetry. The NACTs (τijx and τijy ; i,j = 1−5; i < j) between the five states in the given nuclear configuration space are calculated by employing Coupled-Perturbed Multi Configuration Space SelfConsistent Field (CP-MCSCF) method implemented in MOLPRO. In Section 4.1 we discuss the important results that have been obtained from the ab initio investigations and the subsequent outcome from adiabatic-to-diabatic transformation on this particular system. 3.2. Bz+ Radical Cation. We also have performed ab initio calculations for the benzene radical cation (Bz+) employing two pairwise normal mode coordinate planes, namely, Q16x−Q16y

Q 3x = ρ cos ϕ and Q 3y = ρ sin ϕ

where Q3x, Q3y, and ρ are in dimensionless unit. The five electronic states are labeled as X̃ 2A′2 → 1; Ã 2E″ → 2,3; B̃ 2E′ → 4,5. In this nuclear configuration space, the following nonadiabatic interactions are prominent:61 (i) Strong JT interactions within E″ as well as E′ state leading to 2−3 and 4−5 CIs at the equilibrium D3h point; (ii) Accidental 4−5 CIs between two sheets of E′ state at three equivalent C2v points; (iii) Strong PJT interactions between ground state (A2′ ) and the two sheets of E′ state designated as 1−4 and 1−5 couplings. Figure 1 exhibits the one-dimensional (1D) cuts for the adiabatic PESs for the five doublet states (A′2, E″, E′) of this radical along ρ coordinate at a given ϕ (=30°). Other than the 2−3 and 4−5 JT CIs at the D3h point, the 4−5 CI occurs at ρ ≈ 3.0 and ϕ = 30°, 150°, 270°, which correspond to C2v geometries. This is evident from the plots of the NACTs given in Figure 2. Figure 2a depicts the phi (ϕ) component of 2−3 NACT (τ23 ϕ ) corresponding to the 2−3 JT CI in ρ−ϕ coordinate space. Figure 2b shows the plot of phi (ϕ) component of 4−5 NACT (τ45 ϕ ) in ρ−ϕ space that represents 4−5 JT CI at ρ → 0 as well as 4−5 C2v CIs at ρ ≈ 3.0 and ϕ = 30°, 150°, 270° in the form of singularities. Finally, Figure 2c and Figure 2d exhibit the plots of phi (ϕ) component of 1−4 and 15 1−5 NACTs (τ14 ϕ and τϕ ), respectively, in the same polar plane. The functional forms of these two τ in this configuration space represent the signatures of 1−4 and 1−5 PJT interactions. E


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We have substituted the ab initio calculated NACTs in the ADT equations [eq 16], and those stiff differential equations are solved over a two-dimensional (2D) grid of geometries by using backward differential formula with appropriate relative and absolute error tolerances to achieve convergence. The 10 coupled ADT equations are solved in polar coordinates (ρ, ϕ) over the ranges from ρ = 0 to 5.0 and from ϕ = 0 to 2π spanning 50 × 180 grid (ab initio) points. One can integrate two sets of coupled differential equations using infinite number of different chosen paths, and each path is supposed to produce a different set of ADT angles, where those angles are expected to show gauge invariance. While the differential equations are integrated for ADT angles along the ϕ grid from 0 to 2π for each positive step of integration from ρ = 0.0 to 5.0, such angles turn into integer multiple of π or zero if the corresponding NACTs do or do not show singularity at certain point(s) enclosed by the contour. Figure 3 represents 1D cuts of four ADT angles along ϕ coordinate at ρ = 5.0, where at the end of closing the contour (ϕ = 2π): (a) θ23 depicts the presence of one JT CI within 2−3 states by attending the value π; (b) θ45 validates one JT CI and three accidental CIs within 4−5 states by its magnitude 4π; and (c) θ14 and θ15 attain the value of zero (0) even though those angles show nonzero values at intermediate ϕ values. In order to take the advantage of inherent symmetry of the molecule (D3h), we solve the ADT equations for the chosen pair

Figure 1. 1D cuts of adiabatic PESs of NO3 radical for Q3x−Q3y pairwise modes along ρ at ϕ = 30°, where ρ and ϕ are the polar coordinates corresponding to the normal coordinates of degenerate asymmetric stretching mode. ρ is in dimensionless unit. The 2−3 and 4−5 CIs occur at D3h point, where at C2v points, the 4−5 accidental CIs are present at ϕ = 30°, 150°, 270° and ρ ≈ 3.0.

Figure 2. NACTs of NO3 radical for Q3x−Q3y pairwise modes along the polar coordinates ρ and ϕ: (a) τ23 ϕ illustrates the JT CI within E″ electronic state (2−3) at D3h geometry (ρ = 0). (b) τ45 ϕ represents the JT CI within E′ state (4−5) at D3h point (ρ = 0) and three equivalent CIs at C2v points. 15 (c) τ14 ϕ and (d)τϕ depict PJT interactions between lowest A2′ state and excited E′ state (1−4 and 1−5, respectively). F


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transformation matrix, and the observables are expected to be same irrespective of the path.37 While integrating the ρ coordinate from 5.0 to 0.0 for each positive step integration of ϕ grid from 0 to 2π, it appears that out of 10 possible ADT angles, four are significant ones. Figure 4 represents the functional forms of those ADT angles (θij) plotted in ρ−ϕ plane (2D) corresponding to Q3x−Q3y configuration space. It is evident from Figure 4a and Figure 4b that θ23 and θ45 generated from the newly constructed path also carry the pronounced characteristics of CIs (JT as well as accidental). Moreover, Figure 4c and Figure 4d depict the ADT angles θ14 and θ15 corresponding to the 1−4 and 1−5 PJT interactions, respectively. The ADT matrices are calculated by plugging those angles (θ) in eq 15, where the diagonal elements of ADT matrix, A22(=A33) and A44(=A55) clearly show sign changes as a function of ϕ in Figure 5a and Figure 5b validating the presence of 2−3 and 4−5 CIs. According to Longuet-Higgins, BO adiabatic eigenfunctions flip their sign along a closed loop surrounding a conical intersection. In other words, the diagonal elements of A matrix corresponding to that wave function also change their sign to maintain the single-valuedness of diabatic wave function. Finally, the diabatic potential matrix W is evaluated from eq 11 using the obtained ADT matrix. Figure 6a and Figure 6b show the fourth and fifth diabatic PESs, respectively, whereas Figure 6c represents the coupling element between those two states. It is noteworthy to mention that the calculated diabatic potential energy matrix elements are found to be smooth, symmetric, and continuous in the given nuclear configuration space.

Figure 3. 1D cuts of the ADT angles of NO3 radical for Q3x−Q3y pairwise modes along ϕ coordinate for a particular value of ρ(=5.0). θ23 attains the magnitude of π indicating only one JT CI between 2−3 states, and θ45 reaches the value of 4π at the end of a closed contour representing four CIs (one JT CI and three accidental CIs) within 4−5 states. θ14 and θ15 acquire the initial value at ϕ = 0° after closing the contour at ϕ = 2π.

of normal modes (Q3x−Q3y) along the ρ coordinate for each ϕ grid, obtain the ADT angles, and construct the diabatic PESs. It may be noted, whatever be the path chosen, the resulting diabatic PESs will be related through an orthogonal

Figure 4. ADT angles of NO3 radical for Q3x−Q3y pairwise modes along polar coordinates. (a) θ23 and (b) θ45 plotted against ρ and ϕ attain the magnitude of π indicating the CIs between 2−3 and 4−5 states, respectively. (c, d) The functional form of the ADT angles θ14 and θ15 respectively, for the PJT interactions. G


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The Journal of Physical Chemistry A following relations: Q 16x = ρ cos ϕ and Q 16y = ρ sin ϕ

where Q16x, Q16y, and ρ are in dimensionless unit. The five electronic states of Bz+ are labeled as E1g → 1,2; E2g → 3,4; A2u → 5. Since Q16x and Q16y are JT active component modes, we observe the strong JT symmetry breaking phenomena within E1g as well as E2g states represented as 1−2 and 3−4 JT interaction. As a result, we expect singularities in the functional 34 form of the NACTs, τ12 ϕ and τϕ . Figure 7a and Figure 7c display 1D cuts of adiabatic PESs of E1g and E2g states of Bz+, respectively, plotted along ρ coordinate at a given ϕ (=45°) in 34 Q16x−Q16y plane. The corresponding NACTs, τ12 ϕ and τϕ , for those JT CIs are plotted Figure 7b and Figure 7d, respectively. From the functional form of the NACTs, it is evident that at 34 ρ → 0.0, τ12 ϕ and τϕ tend to blow up. To depict the functional form of those NACTs on 2D plane (ρ−ϕ space), we also incorporate Figure 8a and Figure 8b. The ab initio calculated NACTs are then used to solve the 10 coupled ADT equations over 2D grid of geometries over the ranges from ρ = 0 to 2.0 and from ϕ = 0 to 2π spanning 20 × 180 grid (ab initio) points by the same numerical technique with appropriate relative and absolute error tolerances. The integration is first performed along ϕ coordinate for each positive increment from ρ = 0.0 to ρ = 2.0, and the resulting ADT angles, that is, θ12 and θ34, are plotted at ρ = 2.0 in Figure 9. Those angles attend the magnitude of π bearing the signature of JT CIs within 1−2 and 3−4 electronic states. While constructing the diabatic PESs, we choose the path of integration along ρ grid for each positive increment in ϕ coordinate to get the ADT angles as well as to construct the ADT matrices and diabatic PESs. Figure 10a and Figure 10c represent the ADT angles θ12 and θ34, respectively, in ρ−ϕ space for Q16x−Q16y plane, and both of them clearly verify the JT CIs at the limiting value of ρ = 0. The diagonal elements of A matrix, namely, A11 (=A22) and A33 (=A44), plotted in Figure 10b and Figure 10d show necessary sign changes indicating the CIs. The ADT matrix elements are then used in eq 11 to calculate the diabatic potential energy matrix. Figure 11a and Figure 11b represent the 1−2 and 3−4 coupling, respectively, which appear to be smooth and continuous. 4.2.2. The Q19−Q16x Plane. Q19 mode is one of the two components of degenerate out-of-plane CH bend mode (ν19) of Bz+. This mode is mainly responsible for the interstate coupling of 4−5 states of Bz+ in various regions of nuclear configuration space. In particular, we have chosen Q19−Q16x plane to study the 4−5 accidental CI occurring due to the Q19 mode.

Figure 5. Diagonal elements of A matrix of NO3 radical for Q3x−Q3y pairwise modes. (a) A22 plotted in ρ−ϕ grid changes sign representing the JT CI within 2E″ state, and similarly (b) A44 indicates CIs within 2 E′ state.

4.2. Bz+ Radical Cation. Two different nuclear configuration spaces made of three vibrational component modes of Bz+ radical cation are chosen for performing the adiabatic-todiabatic transformation, specifically, Q16x−Q16y and Q19−Q16x planes. We discuss the findings in each pairwise mode in the following subsections. 4.2.1. The Q16x−Q16y Plane. The Q16x and Q16y modes are the two degenerate components of the asymmetric stretching Jahn−Teller (JT) active vibrational mode (ν16). The two components are transformed into the polar form according to the

Figure 6. For Q3x−Q3y pair of NO3 radical, diabatic PESs (a) W44 and (b) W55 are plotted in ρ−ϕ plane and diabatic coupling element within 2 E′ state; i.e., (c) W45, which is smooth, continuous and single-valued, is also plotted in the same plane. H


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Figure 7. (upper) 1D cuts of adiabatic PESs of Bz+ at ϕ = 45° in Q16x−Q16y plane for (a) E1g and (c) E2g electronic states and (e) at ϕ = 90° in 34 Q19−Q16x plane for upper sheet of the E2g state and A2u state along ρ. (lower) 1D cuts of the corresponding NACTs, i.e., (b) τ12 ϕ and (d) τϕ at at ϕ = 90° in Q −Q plane. All of them are plotted along ρ at a definite ϕ. ϕ = 45° in Q16x−Q16y plane and (f) τ45 ρ 19 16x

were performed on quantum dynamics, where the inclusion of appropriate number of coupled electronic states to constitute a sub-Hilbert space is highly relevant to derive the ADT equations and expressions of τ. In the context of non-adiabatic chemistry, ADT equations for two,6,14 three,35−39 and four34 states have already been deduced to construct diabatic PESs for several real molecular systems on which dynamics has been performed thoroughly with utmost care. On the other hand, various molecules still exist where more than four electronic states are highly coupled, and such coupling must be taken into account while understanding their spectral features, scattering cross section, and also other experimental quantities. In this article, explicit expressions of ADT angles as well as NACTs in terms of those angles for five-state subHilbert space are derived for the first time, which can fulfill the requirement to construct the diabatic surfaces and to calculate the experimental observables. ADT condition, which is used for this derivation, is assumed to be valid for the given sub-Hilbert space, but the applicability of those equations for ADT angles on real systems needs to be properly justified. For this reason, two molecular species are taken into account, where the nonadiabatic interactions between the first five electronic states are notably strong. One of them is nitrate radical (NO3), whose symmetry at the equilibrium geometry was ambiguous even up to recent times55 due to so-called strong PJT interaction, and another is benzene radical cation (Bz+), whose JT and PJT interactions along with accidental CIs drew attention to the community over the past few decades.

As usual these two components are transformed into polar counterparts: Q 19 = ρ cos ϕ and Q 16x = ρ sin ϕ

Figure 7e shows the 1D adiabatic cuts for the upper sheet of E2g state and the A2u state plotted along ρ for ϕ = 90°. These two surfaces undergo CI at geometry with D2h symmetry. The τ45 ρ is plotted at ϕ = 90° in Figure 7f showing the singularity corresponding to the 4−5 CI, where this NACT is shown in 2D plane (ρ−ϕ space) in Figure 8c to highlight the position of CI. Likewise the previous cases, numerical integration for ADT angles is pursued with the ab initio calculated NACTs using two different paths. During ϕ integration, ADT angle (θ45) presented at ρ = 2.5 in Figure 9 acquires the numerical value of π at ϕ = 2π verifying accidental CI. On the other hand, while integrating along ρ coordinate, the resulting ADT angle (Figure 10e) also shows prominent non-adiabatic interaction between the 4−5 states, and with those angles, ADT matrix is formed. The diagonal element of the A matrix, that is, A44 (=A55) plotted in Figure 10f shows the required sign changes and validates the presence of 4−5 CI. Finally, for the same configuration space, the diabatic matrix elements are evaluated using the calculated ADT matrix (eq 11), of which the 4−5 coupling term is displayed in Figure 11c, and it appears smooth as well as continuous.

5. CONCLUDING REMARKS So far several theoretical developments took place on Beyond Born−Oppenheimer (BBO) theories, and many calculations I


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Figure 9. ADT angles along ϕ coordinate at ρ = 2.0 for Q16x−Q16y pair of Bz+. θ12 and θ34 reach the value of π due to the presence of JT CIs within 1−2 and 3−4 states, respectively, at the closed contour. θ45 at ρ = 2.5 for Q19−Q16x pair represents the presence of the accidental CI within 4−5 electronic states.

34 Figure 8. NACTs of Bz+ are plotted in ρ−ϕ plane. (a, b) τ12 ϕ and τϕ , for Q −Q plane. respectively, for Q16x−Q16y pair. (c) τ45 ρ 19 16x

In Section 2 of this article, the basic theories of BBO studies are revisited, and the form of explicit expressions for ADT equations and NACTs for five-state sub-Hilbert space are presented for the first time. In the subsequent sections, the workability of those five-state ADT equations are explored with the two aforementioned molecular/ionic systems by carrying out ab initio calculations of five electronic states for their adiabatic PESs and NACTs (Section 3) and then performing the required ADT to construct five-sheeted diabatic surfaces for both the systems (Section 4). From the ab initio study, it is revealed that NO3 possesses high degree of vibronic interactions, which include not only strong JT coupling within its Ã 2E″ (2−3) as well as B̃ 2E′ (4−5) states but also accidental

Figure 10. For Q16x−Q16y pair of Bz+, (a) θ12 and (c) θ34 plotted in ρ−ϕ plane attain the value of π. Similarly for Q19−Q16x pair, (e) θ45 reaches the magnitude of π. For Q16x−Q16y pair, (b) A11 and (d) A33 represent first and third diagonal elements of the A matrix and for Q19−Q16x pair; (f) A44 depicts the fourth diagonal element of A matrix.

CIs between 4−5 states along with appreciable PJT effect on the ground X̃ 2A2′ by the excited B̃ 2E′ state (1−4 and 1−5 interactions). It is worthwhile to mention that majority of these vibronic interactions are being studied by many other groups in order to depict mainly the correct spectral features of this system. Stanton et al.,53,54 Eisfeld et al.,55 and Köppel et al.56,57 J


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validate the theory presented in this article. Though for the present system, the diabatic PESs are always constructed only around the Franck−Condon region for comparison with spectroscopic measurements, the BBO theory is recently shown equally applicable in accurate manner at the asymptotic region for constructing diabatic PESs.39 Benzene radical cation (Bz+) is chosen as another system to validate the workability of the formulated BBO equations. This cation is known for strong non-adiabatic interactions owing to JT and PJT couplings.62,63,65 Many multistate multimode calculations have been done on this system extensively to understand and envisage the photoelectron spectra of benzene molecule.62,66 Nevertheless, in this work, an ab initio based fivestate diabatic Hamiltonian is constructed from first principle for the first time using the newly derived ADT equations. For this purpose, we have chosen two pairwise normal mode coordinate planes as the configuration spaces to explore a few of the major vibronic couplings prevailing within the five states of Bz+. Those are, namely, the JT CIs within E1g (1−2) as well as E2g (3−4) states in Q16x−Q16y pairwise mode and an accidental CI between the upper sheet of E2g state and A2u state (4−5) in Q19−Q16x pairwise mode. Even though the JT CIs are wellknown and have been studied deeply, here in this work the accidental CI along with the JT ones are treated with the BBO formalism for the first time to verify as well as validate their presence in the considered nuclear configuration spaces. On application of the five-state ADT equations, the obtained ADT angles show signature of those CIs as the magnitudes of θ12, θ34, and θ45 become equal to π. Thereby, the diagonal elements of ADT matrix show sign changes validating the CIs. Finally, with the ADT matrix elements, a five-sheeted diabatic Hamiltonian is constructed, where off-diagonal elements are prominent, smooth, and continuous. In this context, we need to stress the fact that this study shows the workability of the newly developed ADT equations, where the adiabatic PESs and NACTs are computed at CASSCF level of theory. In near future, with the help of the present theory, we intend to construct diabatic surfaces using more accurate adiabatic PESs and NACTs (at the MRCI level) to carry out dynamics and simulate experimentally observed photoelectron spectra available for both the systems.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b04592. Elements of Ω matrix, differential equations for ADT angles, and explicit expressions of NACTs (PDF)

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Figure 11. For Q16x−Q16y pair of Bz+, (a) W12 and (b) W34 represent diabatic coupling terms and similarly; for Q19−Q16x pair, (c) W45 depicts the coupling term, which are smooth, continuous, and singlevalued.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

all have led investigations on this radical system and proposed the form of diabatic Hamiltonian, whereas in this work, fivestate diabatization is performed using the newly developed first principle based ADT equations encapsulating all possible non-adiabatic interactions present in NO3 including the 4−5 accidental CIs. The calculated ADT angles attain a magnitude of π whenever a CI is encountered, be it a symmetry required or an accidental type. Indeed, our diabatic PESs and couplings are smooth and symmetric as well as continuous that collectively

Satrajit Adhikari: 0000-0002-2462-4892 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.M. (File No: SPM-07/080(0250)/2016-EMR-I) and B.M. (File No: 09/080(0960)/2014-EMR-I) acknowledge CSIR, India for research fellowship. S.A. acknowledges DST, India, through K


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Project No. EMR/2015/001314 for the research funding and also thanks IACS for CRAY supercomputing facility.

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