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ArticleOctober 3, 2018

Topological Effects in Vibronically Coupled Degenerate Electronic States: A Case Study on Nitrate and Benzene Radical Cation
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Soumya Mukherjee
Soumya Mukherjee
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
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Bijit Mukherjee
Bijit Mukherjee
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
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Joy Dutta
Joy Dutta
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
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Subhankar Sardar
Subhankar Sardar
Department of Chemistry, Bhatter College, Dantan, Paschim Medinipur, 721426, India
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Satrajit Adhikari*
Satrajit Adhikari
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
*E-mail: [email protected] (S.A.).
More by Satrajit Adhikari
http://orcid.org/0000-0002-2462-4892

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Cite this: ACS Omega 2018, 3, 10, 12465–12475

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https://doi.org/10.1021/acsomega.8b01648

Published October 3, 2018

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Abstract

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We carry out detailed investigation for topological effects of two molecular systems, NO₃ radical and C₆H₆⁺ (Bz⁺) radical cation, where the dressed adiabatic, dressed diabatic, and adiabatic-via-dressed diabatic potential energy curves (PECs) are generated employing ab initio calculated adiabatic and diabatic potential energy surfaces (PESs). We have implemented beyond Born–Oppenheimer (BBO) theory for constructing smooth, single-valued, and continuous diabatic PESs for five coupled electronic states [J. Phys. Chem. A2017,121, 6314–6326]. In the case of NO₃ radical, the nonadiabatic coupling terms (NACTs) among the low-lying five electronic states, namely, X̃²A₂^′ (1²B₂), Ã²E″ (1²A₂ and 1²B₁), and B̃²E′ (1²A₁ and 2²B₂), bear the signature of Jahn–Teller (JT) interactions, pseudo JT (PJT) interactions, and accidental conical intersections (CIs). Similarly, Bz⁺ radical cation also exhibits JT, PJT, and accidental CIs in the interested domain of nuclear configuration space. In order to generate dressed PECs, two components of degenerate in-plane asymmetric stretching modes are selectively chosen for both the molecular species (Q_3x–Q_3y pair for NO₃ radical and Q_16x–Q_16y pair for Bz⁺ radical cation). The JT coupling between the electronic states is essentially originated through the asymmetric stretching normal mode pair, where the coupling elements exhibit symmetric and nonlinear functional behavior along Q_3x and Q_16x normal modes.

Subjects

1. Introduction

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The Born–Oppenheimer (BO) approximation (1,2) completely decouples the mechanics of fast-moving electrons and slow-moving nuclei, leading to satisfactory explanation of several experimental findings. On the contrary, complete or partial failure of this theoretical framework could be encountered for excited-state molecular processes such as charge-transfer reactions, measurements of scattering cross sections, and photochemical reactions. (3,4) In such phenomena, the non-negligible electron–nuclear coupling designated as nonadiabatic coupling terms (NACTs) not only plays a significant role for this discrepancy but also attains singular characteristics at degenerate points in the nuclear configuration space (CS). Therefore, overlooking of NACTs, (5,6) one of the major outcomes of BO treatment often leads to erroneous dynamical results and inaccurate numerical data regarding the molecular processes.

In the early sixties, Longuet-Higgins noticed multivaluedness of the eigenfunctions (7) while traversing along a closed path encircling a point of conical intersection (CI). Though this unusual behavior was circumvented by incorporating a phase factor (8) in an ad hoc manner, Mead and Truhlar (9) eliminated this multivaluedness behavior of wavefunctions by encompassing a vector potential into the nuclear Hamiltonian. On the other hand, the Hellmann–Feynman theorem predicts the possibility of singularity in NACTs (10,11) over the nuclear CS whenever electronic states appear to be degenerate or near degenerate. This singular characteristics can be removed by transforming to a different representation or basis, known as the diabatic representation where the singular kinetic coupling terms are converted into smoothly varying off-diagonal terms in the diabatic potential energy matrix. Such unitary transformation is known as the adiabatic-to-diabatic transformation (ADT). In this context, it may be noted that vector field created by the NACTs can be decomposed into longitudinal and transverse components, 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 (the removable part) can be eliminated, (5,12) whereas the transverse or the nonremovable part may be neglected at the close vicinity of a CI.

Hobey and Mclachlan devised the first technique of ADT only for one degree of freedom and latter on, F. T. Smith implemented this method for a diatomic molecule. M. Baer proposed a general formalism for the diabatization of two coupled adiabatic potential energy surfaces (PESs) involved on the process of triatomic collision. In that work, the ADT condition is depicted which is the key mechanistic element for the determination of the ADT matrix elements by integrating the differential equations along a two-dimensional contour over the nuclear CS. (5,12) On the other hand, only the fulfillment of “curl condition” (12,13) guarantees the existence and the uniqueness of the solution to those equations. Generally, ADT angles can be calculated by integrating the NACTs around the CI point(s) and because of the quantized nature of the closed contour integrals, resulting angles will attain magnitudes in multiples of π (pi). (14) The satisfaction of aforementioned curl conditions also validates the presence of a set of electronic states forming a complete space. Though the above said condition is invalid at the degenerate CI point(s), nevertheless, the singular NACTs can be taken out from the SE if the line integrals exhibit quantization within the sub-Hilbert space constituted with all coupled electronic states. (15) The beyond BO (BBO) treatment for three, four, and five coupled electronic states in terms of electronic basis functions or the ADT matrix elements has been generalized by Adhikari et al., (16−19) and the explicit expressions of the NACTs, curl–divergence equations, curl conditions, and the diabatic PESs have also been formulated in terms of the ADT angles. Moreover, this theoretical development has been successfully employed on model (16−18,20) as well as realistic (19,21−24) systems for constructing smooth, continuous, and single-valued diabatic PESs. Molecular systems having strong vibronic interactions between five electronic states are omnipresent in nature. NO₃ radical and C₆H₆⁺ (Bz⁺) radical cation are two highly intricate examples of five electronic state sub-Hilbert spaces. Recently, we have pursued formulation of BBO equations for five state sub-Hilbert spaces, devised the ADT equations and explicit expressions of NACTs in terms of ADT angles, and successfully implemented the formalism for NO₃ radical and Bz⁺ radical cation.

In this article, our aim is to generate dressed adiabatic potential energy curves (PECs), dressed diabatic PECs, and adiabatic PECs via dressed diabatic potentials in a pair of normal mode coordinates for both the molecular/ionic species. The PECs carry the signature of topological effects in their functional variation along a particular normal mode coordinate by taking average over the other normal mode coordinate. Symmetric variation of dressed diabatic couplings indicates the absence of linear and higher odd order functions in electron–nuclear couplings, whereas asymmetric diabatic couplings incorporate both odd and even order polynomials in their functional forms. Detailed topological studies clearly reveal effects of NACTs in stabilization or destabilization of PESs.

NO₃ radical is one of the most challenging molecular species in the realm of nonadiabatic dynamics. It exhibits highly complex spectral features due to intense vibronic coupling between its low-lying five electronic states X̃²A₂^′ (1²B₂), Ã²E″ (1²A₂ and 1²B₁), and B̃²E′ (1²A₁ and 2²B₂) in the Franck–Condon region of nuclear CS. Several theoretical predictions were already proposed to elucidate the underlying principle of the complex spectral envelop, obtained from various experimental tools such as dispersed fluorescence spectroscopy, (25,26) Fourier transform infrared spectroscopy, (27,28) photoelectron spectroscopy, (29) and also from cavity ringdown spectroscopy. (30,31) In spite of various controversies about the ground state equilibrium geometry, Eisfeld and Morokuma (32) finally confirmed the radical to be of D_3h symmetry. Mayer et al., (33) Faraji et al., (34) and Stanton et al. (35,36) also proposed some theoretical development to reveal the key features of the experimentally obtained spectral profile. It is noteworthy to mention that strong Jahn–Teller (JT) interactions prevail within the Ã²E″ and B̃²E′ states at the equilibrium D_3h point along with several accidental CIs at C_2v geometries. (19,37) In addition, the ground state, X̃²A₂^′ experiences strong pseudo JT (PJT) interaction due to the B̃²E′ state. (19,37) These vibronic phenomena strongly influence the nuclear dynamics and the associated features of the photoelectron spectra (29,38) of the NO^3– anion.

The photophysics of Bz⁺ radical cation is one of the center of attraction in the arena of BBO theory. In all theoretical developments, (39−47) the electronic states, namely, X̃²E_1g, B̃²E_2g, C̃²A_2u, D̃²E_1u, and Ẽ²B_2u, along with the CIs among the PESs were investigated for its dynamical studies. Goode et al. (48) pointed out the complex vibrational pattern of B̃²E_2g state using the combined effect of Herzberg–Teller and PJT coupling between this state and close-lying C̃²A_2u state. JT coupling parameters for e_2g vibrational modes of this radical cation which are used to illustrate the photoinduced Rydberg ionization spectra of the B̃²E_2g state (49) were extensively measured by Johnson. (50) The low-lying doublet states of Bz⁺, B̃²E_2g, and C̃²A_2u show intense nonadiabatic interactions (39) among them. The JT effects within the ground, X̃²E_1g as well as the excited state, B̃²E_2g are quite predominant particularly lowering the symmetry of ground state equilibrium structure of Bz⁺ to D_2h. (42,50) Apart from the JT effects, there are accidental CIs and PJT interactions prevailing between B̃²E_2g and C̃²A_2u states. (39) We have investigated the JT interactions of the ground (X̃²E_1g) and excited states (B̃²E_2g) as well as accidental CIs within the higher excited states (B̃²E_2g–C̃²A_2u) using the ADT equations by employing the BBO formalism. (19)

The article is arranged as follows: Section 2 presents detailed theoretical framework of BBO theory for a five electronic state sub-Hilbert spaces. This section also gives a brief description of the underlying equations for calculating dressed potentials. In Section 3, we explore topological effects prevailing in both of the molecular systems. This detailed investigation will be highly useful to predict the possible functional forms of the inherent nonadiabatic couplings. Finally, the summary of this topological study is laid out in Section 4.

2. BBO Theory and Diabatic Hamiltonian

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As per the BO treatment, total molecular wavefunction can be expanded as a linear combination of the electronic wavefunctions where the combining coefficients represent the nuclear counterpart. While considering a sub-Hilbert space of finite dimension, a total electron–nuclei wavefunction takes the following form

(1)

This wavefunction is the eigenfunction of molecular SE, and the corresponding eigenvalue represents the total energy of the system

(2)

The molecular Hamiltonian can be partitioned into a nuclear kinetic energy operator (T̂_n) and the electronic Hamiltonian (Ĥ_e(s_e|s_n))

(3)

where the nuclear kinetic energy operator is defined as

(4)

On the other hand, electronic wavefunctions (|ξ_i(s_e|s_n)⟩), that is, the basis of the BO expansion, are eigenfunctions of Ĥ_e(s_e|s_n) operator with a nuclear coordinate-dependent eigenvalue u_i(s_n). Therefore, the electronic eigenvalue equation is presented as below

(5)

When the BO expansion of molecular wavefunction is substituted in total Hamiltonian [eq 2] and it is projected with various electronic wavefunctions, the compact adiabatic (kinetically coupled) SE takes the following matrix form

(6)

where U_ij = u_iδ_ij and τ⃗ is the nonadiabatic coupling matrix (NACM) defined as

(7)

The adiabatic representation of the SE is difficult to solve because of the numerical instability of the NACTs at the points of degeneracy. Therefore, we encounter the limitations of the adiabatic representations which exhibit the inevitable necessity of a different representation where the nonadiabatic interactions appear as continuous, smooth, and single-valued diabatic coupling terms. This transformation could be achieved by using an orthogonal rotation matrix (A)

(8)

where ψ and ϕ^d symbolize the adiabatic and diabatic nuclear wavefunction, respectively.

While incorporating the aforementioned form of wavefunction, diabatic representation of the nuclear SE takes the following expression

(9)

where

(10)

under the condition

(11)

known as the ADT condition.

The ADT condition [eq 11] can be used as the precursor of devising the ADT equations. For five (5) electronic state sub-Hilbert spaces, the model ADT matrix can be constructed by multiplying ⁵C₂ rotation matrices in a particular way. Those could be arranged in ⁵C₂! different ways, but we carry out the formulation for a particular order of multiplication. On the other hand, anyone of the possible arrangements produces same numerical value of the diabatic potential energy matrices. For any one of the rotation matrices, A_ij(θ_ij), where ii and jj elements become cosθ_ij and ij and ji elements take the form of sinθ_ij and −sinθ_ij, respectively. Among the remaining elements, diagonal terms are one, whereas the off-diagonal terms acquire the magnitude of zero.

For five state sub-Hilbert space, the order of multiplication is taken as

(12)

When this model rotation matrix and NACM are incorporated in eq 11, a set of differential equations (ADT equations) can be formulated by solving the matrix form of ADT condition. Because of the absence of any analytic solution of these differential equations, they must be integrated numerically to obtain the ADT angles which are used to construct the ADT matrix at each and every grid points in the interested domain of the nuclear CS. During the transformation of adiabatic PESs to the diabatic one, a similarity transformation [eq 10] is carried out such that the NACTs take the form of diabatic coupling elements which appear in diagonal and off-diagonal elements of the diabatic potential energy matrix.

ADT equation for “1–2” coupling of five coupled electronic states can be formulated as (19)

(13)

If the adiabatic PESs are used for the simulation of the photoelectron spectrum of any molecule, the nonadiabatic interactions are overlooked and the inconsistency between the theoretically predicted and experimentally probed spectrum is maximum at the points of JT as well as accidental CIs. Diabatization not only removes the singularity of the NACTs but also can be used for carrying out nuclear dynamics to reproduce the photodetachment spectra (21) and to study the reactive scattering dynamics (51) to obtain integral cross sections.

3. Dressed Adiabatic, Dressed Diabatic, and Adiabatic-via-Dressed Diabatic PECs: Topological Effects

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In a polyatomic molecule, nonadiabatic couplings could exhibit multidimensional functional behavior over the nuclear CS due to the interactions of multiple electronic states through several vibrational degrees of freedom. In order to understand the contribution of each vibrational mode on the overall nonadiabatic interactions in a molecule, detailed investigation of topological effects is the most suitable approach. Moreover, it is readily realizable that functional features of PESs and NACTs defined over a multidimensional vector space cannot be displayed in the Euclidean space because of restriction in the number of independent axes.

In our present ab initio calculation, PESs and NACTs are evaluated as functions of two normal modes (Q_i and Q_j), keeping the other ones at their equilibrium. As a consequence, while constructing dressed potential curves (adiabatic as well as diabatic) and dressed diabatic couplings along a specific coordinate, the average effect of other mode can be embedded by integrating adiabatic and diabatic potential matrix elements over the second one. In this topological investigation, we have employed ab initio-based calculation to evaluate adiabatic PESs and NACTs and use first-principles-based transformation equations to construct diabatic PESs as a function of different pairs of normal modes, where the other modes are at equilibrium for each case. Such adiabatic and diabatic PESs are averaged over ground vibrational eigenfunctions of respective electronic states for a specific mode, and PECs are obtained to realize the topological effect originating from the other mode. The dimensionless mass-weighted normal mode coordinates can be defined as

(14)

where ω_i and m_k signify the vibrational frequency of ith vibrational mode and atomic masses, respectively. On the other hand, x_ki and x_k⁰ represent Cartesian coordinates at distorted and equilibrium nuclear geometry.

The dressed adiabatic potentials (

) for ith mode and pth electronic state can be defined as

(15)

where u_p(Q_j|Q_i)s represent five adiabatic PESs and ζ_p,v=0^a(Q_j|Q_i) signifies ground vibrational eigenfunction (v = 0) of the corresponding adiabatic PESs (p). Here,

are the dressed adiabatic PESs along Q_i where the other effects of other normal mode (Q_j) are averaged out.

Similarly, dressed diabatic potentials (

) for ith normal mode and m,n electronic state can be expressed as

(16)

When this dressed diabatic energy matrix is diagonalized, adiabatic-via-dressed diabatic potential (

) is obtained.

Dressed adiabatic, diabatic PECs, and dressed diabatic couplings depict 1D functional form of potentials as well as couplings, where the effect of another mode is averaged out with respect to ground vibrational wavefunctions of different electronic states. It is needless to say that dressed PECs and diabatic couplings have nothing to do while studying photoelectron spectrum or performing reactive scattering calculation, but those functions can predict a lot of interesting facts about the structural features of the molecule and analytic expressions of diabatic couplings as a function of a specific coordinate. Indeed, such curves are highly useful to speculate nonadiabatic effects arising from a specific mode on the photoelectron spectrum or reactive cross sections of a molecular process. It is noteworthy to mention that not only the normal mode coordinates but also their associated polar counterparts (ρ and ϕ) may be employed to construct dressed PECs. We can definitely perform topological study along these polar coordinates to explore their contribution on nonadiabatic interactions, but the origination of JT couplings (either sole contribution of even order polynomials, or incorporation of both linear and higher order interactions) cannot be illustrated because of the absence of any symmetric/asymmetric variations of PECs along the ρ coordinate. Dressed adiabatic PECs cannot incorporate any effects of nonadiabatic coupling, whereas dressed diabatic potentials and diabatic coupling elements encompass diagonal and off-diagonal electron–nuclear couplings, respectively. While diagonalizing dressed diabatic matrix, effects of off-diagonal couplings are plugged into the newly generated adiabatic-via-dressed diabatic potentials which include both diagonal and off-diagonal nonadiabatic interactions.

4. Results and Discussion

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4.1. NO₃ Radical

In the realm of nonadiabatic chemistry, NO₃ radical appears to be an excellent representative system for five coupled electronic states. Several research groups have carried out thorough investigations to dig out exact equilibrium structure and complex features of photoelectron spectra. In order to achieve more structural insight, dressed potentials can be constructed by averaging out one of the normal mode coordinates. Low-lying five electronic states, namely, X̃²A₂^′ (1²B₂), Ã²E″(1²A₂ and 1²B₁), and B̃²E′ (1²A₁ and 2²B₂), are mainly responsible for the inherent nonadiabatic interactions. Therefore, we employ polar analogue (ρ and ϕ) of two degenerate components of in-plane asymmetric stretching vibrational mode (ν₃) (symbolized as Q_3x–Q_3y plane) to perform the ab initio calculations. This radical has been optimized employing UCCSD(T) (a spin unrestricted coupled cluster method with all single and double excitations and perturbative accounts of triple excitations) level of calculation, and the optimized parameters are presented in Table 1. (37) Adiabatic PESs for five lowest electronic states (19) have been already constructed using state-averaged complete active space self-consistent field (CASSCF) calculations employing 6-31g** basis set and (9e, 8o) configuration active space (CAS). On the other hand, analytic NACTs (19) between the five low-lying electronic states have been obtained by implementing coupled-perturbed multi configuration space self-consistent field (CP-MCSCF) theory. In this nuclear plane, strong JT interactions are present within E″ as well as E′ states, forming “2–3” and “4–5” JT CIs at the equilibrium D_3h point. Three additional accidental CIs can be observed within two sheets of E′ state at three equivalent C_2v points (ρ ≈ 3.0 and ϕ = 30°, 150°, and 270°). Moreover, nonzero functional forms of τ¹⁴ and τ¹⁵ validate strong PJT interactions between ground state (A₂^′) and the two sheets of E′ state labeled as “1–4” and “1–5” couplings.

Table 1. Optimized Geometry Parameters of NO₃ Radical (37) with UCCSD(T) Method and cc-pVDZ Basis Set

parameters	optimization result (UCCSD(T))
bond length [N–O]	1.237 Å
bond angle [∠O–N–O]	120.0°
normal mode frequency (ν₃)	1121 cm^-1
equilibrium energy	–279.8442 a.u.

In order to exploit the inherent symmetry of the molecule (D_3h), ADT equations are solved for the chosen pair of normal modes (Q_3x–Q_3y) along the ρ coordinate for each ϕ grid. (19) Our aim is to construct dressed adiabatic, dressed diabatic, and adiabatic-via-dressed diabatic potentials along Q_3x and Q_3y normal mode coordinates. Dressed adiabatic and dressed diabatic PECs are plotted as functions of Q_i by taking average over the other normal mode coordinate (Q_j), whereas adiabatic-via-dressed diabatic potentials can be generated by diagonalizing the dressed diabatic energy matrix (W^Q_j(Q_i)). Figure 1 depicts symmetric variation of dressed adiabatic, dressed diabatic, and adiabatic-via-dressed diabatic potential energies along Q_3x mode. On the other hand, PECs plotted in Figure 2 exhibit asymmetric variation while scanning along the Q_3y mode. Such a different functional behavior originates if the component of the asymmetric stretching vibration does or does not produce equivalent geometry during positive and negative distortions. It can be observed that three PECs are widely separated in Figure 1a due to strong PJT couplings within ground A₂^′ and E′ states. On the contrary, they almost coincide in Figure 1b and 1c because of weak JT interaction within E″ state. Finally, Figure 1d and 1e depict variation of third and fourth excited electronic states originated from JT and PJT interactions within the given range of Q_3x normal mode coordinates. Again, Figure 2 represents asymmetric functional features of PECs along Q_3y normal modes. It is noteworthy to mention that the Q_3y normal mode generates nonequivalent geometries at both sides of the equilibrium structure (Q_3y = 0). Moreover, dressed diabatic coupling elements are not only smooth, continuous, and single-valued but also exhibit similar functional characteristics of the diabatic potential. Figure 3 illustrates symmetric variation of dressed diabatic coupling terms [

, and

] along Q_3x coordinate where the functional forms of the dressed diabatic coupling elements are constituted only with even power polynomials of Q_3x. Now, it is needless to say that JT interactions prevailing within E″ as well as E′ states do not originate from any linear coupling. Higher order electron-nuclear coupling plays the major role behind these JT interactions. On the other hand, Figure 4 represents different behaviors of electron-nuclear couplings along positive and negative directions of the Q_3y normal mode. They loose their symmetric nature, and the corresponding functional form of coupling incorporates odd power terms.

Figure 1. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–5) are depicted for (a) ground, (b) first excited, (c) second excited, (d) third excited, and (e) fourth excited states of NO₃ radical along Q_3x normal mode taking average over Q_3y. All of the PECs exhibit symmetric variation and nonlinear functional characteristics.

Figure 2. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–5) are represented for (a) ground, (b) first excited, (c) second excited, (d) third excited, and (e) fourth excited electronic states of NO₃ along Q_3y normal mode taking average over Q_3x. Here, the PECs are asymmetric and incorporate linear functionalities.

Figure 3. Dressed diabatic coupling terms of NO₃ radical along Q_3x normal mode: (a) and (b) are smooth, symmetric as well as continuous appeared due to “2–3” and “4–5” JT CIs and three “4–5” accidental CIs. On the other hand, symmetric and single-valued (c) and (d) represent PJT couplings. Therefore, JT activity is originated solely because of nonlinear couplings along the Q_3x mode.

Figure 4. Dressed diabatic coupling terms of NO₃ radical along the Q_3y normal mode. Here, (a) , (b) , (c) , and (d) all are smooth and continuous but the variation is asymmetric. Along Q_3y normal mode, JT coupling contains linear as well as nonlinear functionalities.

4.2. Bz⁺ Radical Cation

Detailed ab initio study has been already carried out to explore the nonadiabatic interactions prevailing within five low-lying electronic states of the Bz⁺ radical cation. Further investigations are necessary to illustrate the origin and functional forms of dressed potentials as well as dressed diabatic coupling elements. In order to reveal those functional features, we have constructed three types of dressed PECs (adiabatic, diabatic, and adiabatic-via-dressed diabatic) and dressed diabatic coupling terms along two components of the degenerate in-plane asymmetric stretching mode (ν₁₆) (Q_16x and Q_16y). This pairwise normal mode is selectively chosen because of its immense importance while depicting major inherent nonadiabaticities present in this radical cation. Before discussing our theoretical findings, a brief summary of the ab initio details (19) should be clearly mentioned for proper understanding of the newly constructed dressed PECs. Those first-principles-based quantum calculations for neutral species reveal the D_6h symmetric structure at the equilibrium geometry, whereas the cationic species attends D_2h point group due to intense JT distortion. In the case of Bz⁺ radical cation, geometry optimization has been carried out implementing B3LYP (Becke, three-parameter, Lee–Yang–Parr) calculation and the optimized parameters are supplied in Table 2. Ab initio adiabatic PESs have been constructed in CASSCF methodology for the lowest five doublet states of Bz⁺, viz, X̃²E_1g (1²B_3g and 1²B_2g), B̃²E_2g (1²A_g and 1²B_1g), and C̃²A_2u (1²B_1u) employing Gaussian basis set 6-31g** with an active space of 29 electrons distributed over 15 orbitals (29e, 15o). Moreover, analytic NACTs are obtained from CP-MCSCF methodology implemented in MOLPRO (52) quantum chemistry software. Because Q_16x and Q_16y produce strong JT interactions within E_1g and E_2g states at the D_6h point (ρ = 0.0), lower sheets get more stabilized and the corresponding NACTs (τ¹² and τ³⁴) tend to blow up (at ρ = 0.0), validating the presence of JT CIs.

Table 2. Optimized Geometry Parameters of Bz⁺ Radical Cation with B3LYP Method and 6-311g(2d,p) Basis Set

parameters	optimization result (B3LYP)
bond length [C–C]	1.382 Å
bond length [C–H]	1.075 Å
bond angle [∠C–C–C]	120.0°
bond angle [∠H–C–C]	120.0°
normal mode frequency (ν₁₆)	1777 cm^-1
equilibrium energy	–230.7616 a.u.

In order to carry out diabatization of PESs, dimensionless mass-weighted normal mode coordinates are first converted into their polar counterparts (ρ and ϕ) to scan the entire adiabatic PESs and the NACTs. Our ultimate objective is to construct dressed adiabatic, dressed diabatic, and adiabatic-via-dressed diabatic potentials along Q_16x and Q_16y normal mode coordinates. Figure 5 displays symmetric functional forms of dressed PECs for four lowest electronic states along the Q_16x normal mode, whereas the PECs exhibit asymmetric behavior along the Q_16y mode (Figure 6). Likewise NO₃ radical, such type of interesting characteristic features are originated because of the formation of equivalent and nonequivalent geometries along Q_16x and Q_16y coordinates, respectively. For the Bz⁺ radical cation, three types of PECs (dressed adiabatic, dressed diabatic, and adiabatic-via-dressed diabatic) along Q_16x are largely separated from each other (Figure 5a–d) because of strong JT couplings. On the other hand, those PECs are exactly superimposed for fifth electronic state because of their noninvolvement in any nonadiabatic interactions along the Q_16x–Q_16y pairwise mode. It is evident from Figure 5 that adiabatic-via-dressed diabatic curves obtained from diagonalization of dressed diabatic potential matrix exhibit one shallow minimum for ground and second excited electronic states (Figure 5a and 5c) and two shallow troughs for first and third excited ones (Figure 5b and 5d). On the other hand, asymmetric functional features of dressed PECs are exhibited along the Q_16y mode (Figure 6) which signify the incorporation of both odd and even power polynomials in the analytic formulation. Functional forms of dressed diabatic coupling elements also display similar trends along the two normal modes. Symmetric variation of the dressed diabatic coupling terms are represented in Figure 7a and 7b, validating the contribution of only even power polynomials for electron–nuclear couplings in the strong symmetry breaking JT interactions. On the other hand, asymmetric dressed diabatic couplings are depicted along Q_16y coordinate in Figure 8a and 8b, where the nonadiabatic interactions incorporate both odd and even power polynomials for couplings. Therefore, it can be concluded that JT interaction along this component (Q_16y) shows both linear and nonlinear dependence.

Figure 5. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–4) are plotted for low-lying four electronic states of Bz⁺ radical cation in diagrams (a–d), respectively, along Q_16x normal mode taking average over the other component, Q_16y. All PECs undergo symmetric variation and include only the even order functional forms.

Figure 6. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–4) are portrayed for lowest four electronic states of Bz⁺ radical cation in diagrams (a–d), respectively, along the Q_16y normal mode by averaging over Q_16x. Here, PECs are asymmetric and incorporate both linear and nonlinear terms.

Figure 7. Dressed diabatic coupling terms are plotted along Q_16x normal mode of the Bz⁺ radical cation. Figures (a,b) depict and , respectively. Both are symmetric, indicating nonlinear “1–2” and “3–4” JT interactions.

Figure 8. Dressed diabatic coupling terms are plotted with the variation of Q_16y normal mode coordinates of the Bz⁺ radical cation. Figures (a,b) describe and , respectively. Along Q_16y normal mode, JT coupling contains linear functions along with nonlinear terms.

5. Conclusions

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For the last few decades, several investigations have been carried out for highly complex scenario in the field of nonadiabatic dynamics of NO₃ radical and Bz⁺ radical cation. In this perspective, an accurate investigation is necessary to explore the topological effects among the pairwise normal mode coordinates. Till now, nonadiabatic couplings have been extensively studied for two to five electronic state sub-Hilbert spaces (19,21−24) to generate highly accurate diabatic PESs. On the contrary, topological studies have been carried out only for two and three coupled electronic states (22−24) which is relevant for predicting the functional forms of dressed diabatic couplings. It is needless to say that in most cases, linear functions of normal mode coordinates are only responsible for JT stabilization, but we have already explored the sole contribution of even power couplings (24) for JT and RT CIs in the case of NO₂ molecule. It has been clearly revealed that selection rule for the electronic transition in NO₂ molecule originates mainly because of even power coupling elements. In this article, dressed adiabatic, dressed diabatic, and adiabatic-via-dressed diabatic electronic states are constructed for two real molecular systems of five electronic state sub-Hilbert space where one can explore the importance of even power coupling modes in JT interaction apart from the traditional ones involving both odd and even power interactions.

In Sections 2 and 3, the basic theoretical developments of BBO methodology as well as detailed background on the formulation of dressed PECs are presented explicitly and in the following section, importance of those PECs for NO₃ radical and Bz⁺ radical cation is investigated. From the previous theoretical investigations, (19,37) it has been observed that NO₃ possesses several kinds of nonadiabatic interactions, namely, JT CI, accidental CI and PJT coupling, whereas the Bz⁺ radical cation is deliberately chosen for its strong JT interaction. NO₃ exhibits JT interaction within E″ as well as E′ states, accidental CIs between two sheets of E′ states along with substantial amount of PJT couplings between “1–4” and “1–5” states. Several groups have performed detailed study on the vibronic couplings to unfold its structural details and complex spectral features, but in this work, topological study for five coupled electronic states is carried out for the first time. Calculated dressed diabatic coupling terms exhibit symmetric and asymmetric functional behaviors along Q_3x and Q_3y modes, respectively though both of the normal modes are responsible for JT coupling. Similar trend is also observed for Q_16x and Q_16y modes of Bz⁺ radical cation. These modes are mainly responsible for JT CIs within E_1g and E_2g electronic states. Q_16x component produces symmetric variation of PECs as well as dressed diabatic couplings, which clearly indicates the signature of higher order JT interactions originated from even power functions of Q_16x coordinate. On the contrary, Q_16y shows profound asymmetry in dressed PECs and dressed diabatic couplings. For the latter one, analytic forms of JT couplings contain both linear and nonlinear polynomials. Finally, we conclude that since this study validates our previous findings not only for lower but also for higher dimensional sub-Hilbert space, the effect of nonadiabaticity on molecular processes is a topological one.

Author Information

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Corresponding Author
- Satrajit Adhikari - Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India; http://orcid.org/0000-0002-2462-4892; Email: [email protected]
Authors
- Soumya Mukherjee - Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
- Bijit Mukherjee - Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
- Joy Dutta - Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
- Subhankar Sardar - Department of Chemistry, Bhatter College, Dantan, Paschim Medinipur, 721426, India
Notes
The authors declare no competing financial interest.

Acknowledgments

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S.M. (file no.: SPM-07/080(0250)/2016-EMR-I) and B.M. (file no.: 09/080(0960)/2014-EMR-I) thank CSIR India for research fellowship, and J.D. acknowledges IACS for the same. S.S. thanks Principal, Bhatter College, Dantan for supplying research facility in his institution. S.A. is thankful to DST, India, through project no. EMR/2015/001314 for research funding. S.A. also acknowledges IACS for CRAY super computing facility.

References

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Paul, A. K.; Sardar, S.; Sarkar, B.; Adhikari, S. Single Surface Beyond Born-Oppenheimer Equation for a Three-State Model Hamiltonian of Na₃ Cluster. J. Chem. Phys. 2009, 131, 124312, DOI: 10.1063/1.3236839
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Mukherjee, S.; Mukherjee, B.; Adhikari, S. Five Electronic State Beyond Born-Oppenheimer Equations and Their Applications to Nitrate and Benzene Radical Cation. J. Phys. Chem. A 2017, 121, 6314– 6326, DOI: 10.1021/acs.jpca.7b04592
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Five Electronic State Beyond Born-Oppenheimer Equations and Their Applications to Nitrate and Benzene Radical Cation
Mukherjee, Soumya; Mukherjee, Bijit; Adhikari, Satrajit
Journal of Physical Chemistry A (2017), 121 (33), 6314-6326CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
We present explicit form of Adiabatic to Diabatic Transformation (ADT) equations and expressions of nonadiabatic coupling terms (NACTs) for a coupled five-state electronic manifold in terms of ADT angles between electronic wavefunctions. ADT matrixes eliminate the numerical instability arising from singularity of NACTs and transform the adiabatic Schrodinger equation to its diabatic form. Two real mol. systems NO3 and C6H6+ (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 [X2A2' (12B2), A2E''(12A2 and 12B1) and B2E'(12A1 and 22B2)], in which all types of non-adiabatic interactions i.e. 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+ [X2E1g (12B3g and 12B2g), B2E2g (12Ag and 12B1g) and C2A2u (12B1u)] depict similar kind of complex feature of non-adiabatic effects. For NO3 radical, the two components of degenerate in-plane asym. 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 asym. stretching mode and the other one is constituted with one of the components each from out-of-plane degenerate bend and in-plane degenerate asym. stretching modes. We calc. ab initio adiabatic potential energy surfaces (PESs) and NACTs among the lowest five electronic states at the CASSCF level using MOLPRO quantum chem. package. Subsequently, the ADT is carried out using those newly developed equations to validate the positions of the CIs, evaluate the ADT angles and construct smooth, sym. and continuous diabatic PESs for both the mol. systems.
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Paul, A. K.; Ray, S.; Mukhopadhyay, D.; Adhikari, S. Ab initio Calculations on the Excited States of Na₃ Cluster to Explore Beyond Born-Oppenheimer Theories: Adiabatic to Diabatic Potential Energy Surfaces and Nuclear Dynamics. J. Chem. Phys. 2011, 135, 034107, DOI: 10.1063/1.3609247
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Ab initio calculations on the excited states of Na3 cluster to explore beyond Born-Oppenheimer theories: Adiabatic to diabatic potential energy surfaces and nuclear dynamics
Paul, Amit Kumar; Ray, Somrita; Mukhopadhyay, Debasis; Adhikari, Satrajit
Journal of Chemical Physics (2011), 135 (3), 034107/1-034107/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We perform ab initio calcn. using quantum chem. package (MOLPRO) on the excited states of Na3 cluster and present the adiabatic PESs for the electronic states 22E' and 12A'1, and the non-adiabatic coupling (NAC) terms among those states. Since the ab initio calcd. NAC elements for the states 22E' and 12A'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 theor. means on beyond Born-Oppenheimer treatment. Nuclear dynamics has been carried out on those diabatic surfaces to reproduce the exptl. spectrum for system B of Na3 cluster and thereby, to explore the numerical validity of the theor. development on beyond Born-Oppenheimer approach for adiabatic to diabatic transformation. (c) 2011 American Institute of Physics.
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Mukherjee, S.; Bandyopadhyay, S.; Paul, A. K.; Adhikari, S. Construction of Diabatic Hamiltonian Matrix from Ab Initio Calculated Molecular Symmetry Adapted Nonadiabatic Coupling Terms and Nuclear Dynamics for the Excited States of Na₃ Cluster. J. Phys. Chem. A 2013, 117, 3475– 3495, DOI: 10.1021/jp311597c
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Construction of Diabatic Hamiltonian Matrix from ab Initio Calculated Molecular Symmetry Adapted Nonadiabatic Coupling Terms and Nuclear Dynamics for the Excited States of Na3 Cluster
Mukherjee, Saikat; Bandyopadhyay, Sudip; Paul, Amit Kumar; Adhikari, Satrajit
Journal of Physical Chemistry A (2013), 117 (16), 3475-3495CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
We present the mol. symmetry (MS) adapted treatment of nonadiabatic coupling terms (NACTs) for the excited electronic states (22E' and 12A1') of Na3 cluster, where the adiabatic potential energy surfaces (PESs) and the NACTs are calcd. at the MRCI level by using an ab initio quantum chem. package (MOLPRO). The signs of the NACTs at each point of the configuration space (CS) are detd. 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 cor. NACTs and the corresponding ADT angles demonstrate the validity of curl condition for the existence of three-state (22E' and 12A1') sub-Hilbert space, it becomes possible to construct the continuous, single-valued, sym., 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 the exptl. spectrum for system B of Na3 cluster.
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Mukherjee, S.; Adhikari, S. The Excited States of K₃ Cluster: The Molecular Symmetry Adapted Non-Adiabatic Coupling Terms and Diabatic Hamiltonian Matrix. Chem. Phys. 2014, 440, 106– 118, DOI: 10.1016/j.chemphys.2014.05.022
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The excited states of K3 cluster: The molecular symmetry adapted non-adiabatic coupling terms and diabatic Hamiltonian matrix
Mukherjee, Saikat; Adhikari, Satrajit
Chemical Physics (2014), 440 (), 106-118CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
We calc. the adiabatic potential energy surfaces (PESs) and the nonadiabatic coupling terms (NACTs) for the excited electronic states of K3 cluster by MRCI approach using MOLPRO. The NACTs are adapted with mol. symmetry to assign appropriate IREPs so that the elements of the Hamiltonian matrix are totally sym. We incorporate those NACTs into three-state adiabatic-to-diabatic transformation (ADT) equations to obtain ADT angles for constructing continuous, single-valued, smooth and sym. diabatic Hamiltonian matrix, where its elements are fitted with analytic functions. Finally, we demonstrate that the dressed diabatic and adiabatic-via-dressed diabatic PECs show prominent topol. effect over dressed adiabatic curves.
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Mukherjee, S.; Mukhopadhyay, D.; Adhikari, S. Conical Intersections and Diabatic Potential Energy Surfaces for the Three Lowest Electronic Singlet States of H₃⁺. J. Chem. Phys. 2014, 141, 204306, DOI: 10.1063/1.4901986
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Conical intersections and diabatic potential energy surfaces for the three lowest electronic singlet states of H3+
Mukherjee, Saikat; Mukhopadhyay, Debasis; Adhikari, Satrajit
Journal of Chemical Physics (2014), 141 (20), 204306/1-204306/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We calc. the adiabatic Potential Energy Surfaces (PESs) and the Non-Adiabatic Coupling Terms (NACTs) for the three lowest singlet states of H3+ in hyperspherical coordinates as functions of hyperangles (θ and .vphi.) for a grid of fixed values of hyperradius (1.5 ≤ ρ ≤ 20 bohrs) using the MRCI level of methodol. employing ab initio quantum chem. package (MOLPRO). The NACT between the ground and the first excited state translates along the seams on the θ - .vphi. space, i.e., there are six Conical Intersections (CIs) at each θ (60° ≤ θ ≤ 90°) within the domain, 0 ≤ .vphi. ≤ 2π. While transforming the adiabatic PESs to the diabatic ones, such surfaces show up six crossings along those seams. Our beyond Born-Oppenheimer approach could incorporate the effect of NACTs accurately and construct single-valued, continuous, smooth, and sym. diabatic PESs. Since the location of CIs and the spatial amplitudes of NACTs are most prominent around ρ = 10 bohrs, generally only those results are depicted. (c) 2014 American Institute of Physics.
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Mukherjee, S.; Mukherjee, B.; Sardar, S.; Adhikari, S. Ab Initio Constructed Diabatic Surfaces of NO₂ and the Photodetachment Spectra of its Anion. J. Chem. Phys. 2015, 143, 244307, DOI: 10.1063/1.4938526
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Ab initio constructed diabatic surfaces of NO2 and the photodetachment spectra of its anion
Mukherjee, Saikat; Mukherjee, Bijit; Sardar, Subhankar; Adhikari, Satrajit
Journal of Chemical Physics (2015), 143 (24), 244307/1-244307/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A thorough investigation has been performed for electronic structure, topol. effect, and nuclear dynamics of NO2 mol., where the adiabatic potential energy surfaces (PESs), conical intersections between the ground (X2A1) and the first excited state (A2B2), and the corresponding non-adiabatic coupling terms between those states are recalcd. [Chem. Phys. 416, 11 (2013)] to achieve enough accuracy in dynamics. We employ beyond Born-Oppenheimer theory for these two state sub-Hilbert space to carry out adiabatic to diabatic transformation (ADT) to obtain the ADT angles and thereby, to construct single-valued, smooth, and continuous diabatic PESs. The analytic expressions for the adiabatic PESs and ADT angles are provided to represent a two-state three-mode diabatic Hamiltonian of NO2 for performing nuclear dynamics to calc. the photo-electron spectra of its anion. It appears that not only Jahn-Teller type coupling but also Renner-Teller interaction contributes significantly on the overall spectrum. The coupling between the electronic states (X2A1 and A2B2) of NO2 is essentially through the asym. stretching mode, where the functional form of such interaction is distinctly sym. and non-linear. (c) 2015 American Institute of Physics.
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Nelson, H. H.; Pasternack, L.; McDonald, J. R. Laser-Induced Excitation and Emission Spectra of Nitrate Radical (NO₃). J. Phys. Chem. 1983, 87, 1286– 1288, DOI: 10.1021/j100231a003
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Laser-induced excitation and emission spectra of nitrate radical (NO3)
Nelson, H. H.; Pasternack, Louise; McDonald, J. R.
Journal of Physical Chemistry (1983), 87 (8), 1286-8CODEN: JPCHAX; ISSN:0022-3654.
The fluorescence excitation and emission spectra of NO3 (2E' (2B2) ↔ 2A2' (2B2)) are reported. Assignments were made for excited- and ground-state vibrational frequencies. A lower limit of 23 μs was measured for the fluorescence lifetime which is at least a factor of 40 longer than predicted from absorption. Spectroscopic observations are consistent with a C2v ground-state geometry.
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Ishiwata, T.; Fujiwara, I.; Naruge, Y.; Obi, K.; Tanaka, I. Study of Nitrate Radical by Laser-Induced Fluorescence. J. Phys. Chem. 1983, 87, 1349– 1352, DOI: 10.1021/j100231a016
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Study of nitrate radical by laser-induced fluorescence
Ishiwata, Takashi; Fujiwara, Ichiro; Naruge, Yukio; Obi, Kinichi; Tanaka, Ikuzo
Journal of Physical Chemistry (1983), 87 (8), 1349-52CODEN: JPCHAX; ISSN:0022-3654.
The nitrate radical, NO3, was directly detected by the laser-induced fluorescence method. The band structure of the fluorescence excitation spectrum was consistent with that of the absorption spectrum. The fluorescence spectrum excited at 662 nm corresponding to the 0-0 transition showed progressions with 1060- and 1480-cm-1 intervals, which were assigned to the sym. stretching (ν1) and degenerate antisym. stretching (ν3) modes of the ground state NO3, resp. The fluorescence lifetime excited at the 0-0 band was estd. to be 2.8 μs from the Stern-Volmer plots in the pressure region of 0.04-0.6 torr.
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Ishiwata, T.; Tanaka, I.; Kawaguchi, K.; Hirota, E. Infrared diode laser spectroscopy of the NO₃ ν₃ band. J. Chem. Phys. 1985, 82, 2196– 2205, DOI: 10.1063/1.448362
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Infrared diode laser spectroscopy of the nitrate radical (NO3)ν3 band
Ishiwata, Takashi; Tanaka, Ikuzo; Kawaguchi, Kentarou; Hirota, Eizi
Journal of Chemical Physics (1985), 82 (5), 2196-205CODEN: JCPSA6; ISSN:0021-9606.
The N-O degenerate stretching band ν3 of the NO3 radical was studied in the gas phase by IR tunable diode laser spectroscopy. The NO3 radical was generated by the reaction of NO2 with an excess of O3. Zeeman modulation was used to observe the paramagnetic absorption lines of 14NO3 and 15NO3 in the wavelength regions 1480-1500 and 1463-1479 cm-1, resp. Only K'' = 3n(n denoting an integer) transitions were obsd., and the N'' = even members were missing from the K'' = 0 manifold. These observations indicate that the NO3 radical belongs to D3h symmetry in the 2A2' ground electronic state. The obsd. spectrum was analyzed by using a sym.-top vibration-rotation Hamiltonian including the spin-rotation interaction. The main parameters thus obtained for 14NO3 are B3 = 0.455 22(11), C3 = 0.227 13(6), Cζ3 = 0.044 79(11), q3 = 0.001 624(33), t3 = 0.000 000 458 0(43), B0 = 0.457 46(12), C0 = B0/2 (fixed), εbb = 0.0280(27), and εcc = 0.1197(36) for v3 = 1, εbb = 0.0277(28), and εcc = 0.1117(34) for v = 0, and ν0 = 1492.3929(9), all in cm-1 with 1 std. error in parentheses. Although these parameters well reproduced the obsd. spectrum, the following anomalous features were noted: (1) a large εcc spin-rotation interaction const. was required to explain the spin splittings for both the ν3 and ground states, (2) a higher-order vibration-rotation interaction term having Δk = ± 4 and Δl = .-+. 2 needed to be included, with the corresponding interaction const. t3 larger than that of CHF3, and (3) the centrifugal distortion consts. and the 1st order Coriolis coupling const. which were derived did not agree with those calcd. assuming a reasonable force field.
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Hirota, E.; Ishiwata, T.; Kawaguchi, K.; Fujitake, M.; Ohashi, N.; Tanaka, I. Near-infrared Band of the Nitrate Radical NO₃ Observed by Diode Laser Spectroscopy. J. Chem. Phys. 1997, 107, 2829– 2838, DOI: 10.1063/1.474641
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Near-infrared band of the nitrate radical NO3 observed by diode laser spectroscopy
Hirota, Eizi; Ishiwata, Takashi; Kawaguchi, Kentarou; Fujitake, Masaharu; Ohashi, Nobukimi; Tanaka, Ikuzo
Journal of Chemical Physics (1997), 107 (8), 2829-2838CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The near-IR band of NO3 obsd. at 7602 cm-1 was analyzed by using diode laser spectroscopy. Most of the spectral lines were recorded using source-frequency modulation. Zeeman modulation was found useful in selectively detecting some Q branch lines, which provided a clue to the assignment of the obsd. spectra. The band satisfied selection rules for a parallel band and was thus ascribed to a 2A1''-2A2' vibronic component assocd. with the 2E''-~X2A2' electronic transition, namely, to a transition from the ground vibronic state to the A1'' vibronic state resulting from excitation of the degenerate in-plane bending mode in the 2E'' electronically excited state manifold. The band was almost free of perturbations, except for some K = 6 lines. The least-squares anal. of 581 assigned lines led to mol. parameters of the upper state, where ground-state parameters were fixed to those obtained from the IR study previously reported. The upper-state B rotational const. gave the effective N-O distance of 1.271 Å, which is to be compared with 1.240 Å in the ground vibronic state. The εbb spin-rotation interaction const. of the upper state was close in magnitude to that in the ground vibronic state, but of opposite sign. This observation indicates that the spin-rotation interaction is primarily caused by that between the 2E'' excited and the ground electronic states.
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Weaver, A.; Arnold, D. W.; Bradforth, S. E.; Neumark, D. M. Examination of the ²A₂′ and ²E′ states of NO₃ by ultraviolet photoelectron spectroscopy of NO₃^–. J. Chem. Phys. 1991, 94, 1740– 1751, DOI: 10.1063/1.459947
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Examination of the 2A2' and 2E'' states of nitrogen oxide (NO3) by ultraviolet photoelectron spectroscopy of the nitrate ion
Weaver, A.; Arnold, D. W.; Bradforth, S. E.; Neumark, D. M.
Journal of Chemical Physics (1991), 94 (3), 1740-51CODEN: JCPSA6; ISSN:0021-9606.
The photoelectron spectrum of the NO3- anion was obtained at 266 and at 213 nm. The 266 nm spectrum probes the 2A2' ground state of NO3. The 213 nm spectrum represents the first observation of the 2E'' lowest-lying excited state of NO3. The 2A2' band shows vibrational progressions in the ν1 sym. stretch and the ν4 degenerate in-plane bend of NO3. The anal. of this band indicates that the NO3 ground state has a D3h equil. geometry and is vibronically coupled to the 2E' second excited state via the ν4 mode. Also obtained was the electron affinity of NO3, 3.937 ± 0.014 eV, and the heat of formation of NO3 at 298 K, 0.777 ± 0.027 eV (17.9 ± 0.6 kcal/mol). The 2E'' state of NO3 lies 0.868 ± 0.014 eV above the ground state. The 2E'' band shows complex and extensive vibrational structure. Several possible assignments of this structure are discussed.
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Deev, A.; Sommar, J.; Okumura, M. Cavity ringdown spectrum of the forbidden Ã²E″ ← X̃²A₂′ transition of NO₃: Evidence for static Jahn-Teller distortion in the Ã state. J. Chem. Phys. 2005, 122, 224305, DOI: 10.1063/1.1897364
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Codd, T.; Chen, M.-W.; Roudjane, M.; Stanton, J. F.; Miller, T. A. Jet cooled cavity ringdown spectroscopy of the Ã²E″ ← X̃²A₂′ transition of the NO₃ radical. J. Chem. Phys. 2015, 142, 184305, DOI: 10.1063/1.4919690
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Jet cooled cavity ringdown spectroscopy of the ~A2E'' ← ~X2A2' transition of the NO3 radical
Codd, Terrance; Chen, Ming-Wei; Roudjane, Mourad; Stanton, John F.; Miller, Terry A.
Journal of Chemical Physics (2015), 142 (18), 184305/1-184305/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The ~A2E←~X2A'2 spectrum of NO3 radical from 7550 cm-1 to 9750 cm-1 was recorded and analyzed. The spectrum differs from previously recorded spectra of this transition due to jet-cooling, which narrows the rotational contours and eliminates spectral interference from hot bands. Assignments of numerous vibronic features can be made based on both band contour and position including the previously unassigned 310 band and several assocd. combination bands. The authors analyzed the spectrum 1st with an independent anharmonic oscillator model and then by a quadratic Jahn-Teller vibronic coupling model. The fit achieved with the quadratic Jahn-Teller model is excellent, but the potential energy surface obtained with the fitted parameters is in only qual. agreement with 1 obtained from ab initio calcns. (c) 2015 American Institute of Physics.
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Eisfeld, W.; Morokuma, K. A Detailed Study on the Symmetry Breaking and its Effect on the Potential Surface of NO₃. J. Chem. Phys. 2000, 113, 5587– 5597, DOI: 10.1063/1.1290607
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A detailed study on the symmetry breaking and its effect on the potential surface of NO3
Eisfeld, Wolfgang; Morokuma, Keiji
Journal of Chemical Physics (2000), 113 (14), 5587-5597CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The tenacious symmetry breaking of the electronic wave function of the nitrate radical (NO3) and its effect on the ground-state potential energy surface is investigated in detail. The symmetry breaking of Hartree-Fock wave functions results from a dominance of the orbital localization effect over the resonance effect and leads to three different solns., one sym. and two distorted ones, for the same electronic state. The resp. equil. geometries of these solns. are points on different potential surfaces, making their comparison meaningless. The resonance effect is promoted by dynamic as well as static electron correlation. However, the dynamic correlation methods [e.g., many-body perturbation theory (MBPT) and coupled-cluster single double (CCSD)] cannot overcome the symmetry breaking of the ref. function and the problem of multiple solns. persists. The symmetry breaking can be avoided by the complete active space SCF (CASSCF) approach that yields unique, single-valued surfaces for all electronic states. However, a sufficiently large and appropriately selected active space has to be used to avoid unphys. distortion of the wave function. Still the orbital localization effect leads to equil. geometries of C2v symmetry which strongly depend on the state-averaging of the CASSCF wave function. Multireference single double CI (MR-SDCI) wave functions are also free of symmetry breaking, if the ref. orbitals are and if the configuration space is invariant under the symmetry operations. MRCI geometry optimizations only result in D3h sym. structures with bond lengths and harmonic frequencies in close agreement with exptl. data.
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Mayer, M.; Cederbaum, L. S.; Köppel, H. Ground State Dynamics of NO₃: Multimode Vibronic Borrowing Including Thermal Effects. J. Chem. Phys. 1994, 100, 899– 911, DOI: 10.1063/1.466572
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Ground state dynamics of NO3: multimode vibronic borrowing including thermal effects
Mayer, M.; Cederbaum, L. S.; Koeppel, H.
Journal of Chemical Physics (1994), 100 (2), 899-911CODEN: JCPSA6; ISSN:0021-9606.
The X2A2' band of the photoelectron spectrum of NO3- is theor. calcd. in the framework of a multimode vibronic coupling model. Linear coupling between and within the X2A2' and the B2E' electronic states of NO3 is the important mechanism governing the dynamics. The necessary coupling consts. are obtained from the ionization potentials of NO3- calcd. at different geometries using ab initio Green's functions. Comparison of the theor. with exptl. results [A. Weaver, D. W. Arnold, S. E. Bradforth, and D. M. Neumark, J. Chem. Phys. 94(3), 1740(1991)] for the PE spectrum shows good agreement if a temp. of the NO3- anion of 455 K in the expt. is assumed. A general theor. treatment of thermal effects in vibronically coupled electronic states is presented. The ground electronic state potential surface of NO3 is discussed in the framework of the vibronic coupling model. A shallow min. at a C2v geometry with a barrier height of 0.006 eV relative to the min. energy D3h configuration is found. It is too weak to deform the effective geometry of the mol. from D3h to C2v. Mode-mode coupling is found to play a relevant role in the ground electronic state of NO3, in general.
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Faraji, S.; Köppel, H.; Eisfeld, W.; Mahapatra, S. Towards a higher-order description of Jahn-Teller coupling effects in molecular spectroscopy: The state of NO₃. Chem. Phys. 2008, 347, 110– 119, DOI: 10.1016/j.chemphys.2007.10.006
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Okumura, M.; Stanton, J. F.; Deev, A.; Sommar, J. New insights into the Jahn-Teller effect in NO₃ via the dark Ã²E″ state. Phys. Scr. 2006, 73, C64– C70, DOI: 10.1088/0031-8949/73/1/n12
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New insights into the Jahn-Teller effect in NO3 via the dark A 2E'' state
Okumura, Mitchio; Stanton, John F.; Deev, Andrei; Sommar, Jonas
Physica Scripta (2006), 73 (1), C64-C70CODEN: PHSTBO; ISSN:0031-8949. (Institute of Physics Publishing)
The recent cavity ringdown (CRD) measurement of the forbidden ∼A 2E'' ← X 2A2' transition of the nitrate radical NO3 reveals a rich, well-resolved spectrum in the near-IR. The spectroscopic detail provides a new window onto the Jahn-Teller (JT) and pseudo-Jahn-Teller (PJT) effects in NO3. This paper reviews the current exptl. evidence for vibronic coupling in the ∼A state and discusses the theor. issues in the context of new preliminary EOMIP/CCSD and CCSD(T) calcns. The theor. results to date indicate that the A 2E'' state of NO3 undergoes a relatively strong JT distortion which may require inclusion of higher order vibronic couplings. The intensity of this transition may involve multiple intensity borrowing mechanisms via PJT coupling among the ∼X, ∼A, and ∼B states.
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Stanton, J. F. On the Vibronic Level Structure in the NO₃ Radical. I. The Ground Electronic State. J. Chem. Phys. 2007, 126, 134309, DOI: 10.1063/1.2715547
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On the vibronic level structure in the NO3 radical. I. The ground electronic state
Stanton, John F.
Journal of Chemical Physics (2007), 126 (13), 134309/1-134309/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The model Hamiltonian approach of Koppel [etal] [Adv. Chem. Phys. 57, 59(1984)] was used to analyze the electronic spectroscopy of the nitrate radical (NO3). Simulations of neg. ion photodetachment of NO3-, the ~X2A'2←~B2E' dispersed fluorescence spectrum of NO3, and the ~B2E←~X2A'2 absorption spectrum are all in qual. agreement with expt., indicating that the model Hamiltonian contains most or all of the essential physics that govern the strongly coupled ~X2A'2 and ~B2E' electronic states of the radical. All 14 bands seen in the dispersed fluorescence spectrum <2600 cm-1 are assigned based on the simulations, filling in a few gaps left by previous work, and 7 addnl. bands <4000 cm-1 are tentatively assigned. The assignment is predicated on the assumption that the ν3 level of NO3 is ∼1000 cm-1 rather than 1492 cm-1 as is presently believed. Support for this reassignment (which assocs. the 1492 cm-1 band with the ν1+ν4 level) comes from both the model Hamiltonian spectrum and a FTIR feature at 2585 cm-1 that is consistent with the large and pos. cross anharmonicity between ν1 and ν4 needed for the alternative 1492 cm-1 assignment. An apparent systematic deficiency exists in the treatment of the model Hamiltonian for levels involving ν4. A discussion of the correlation between energy levels in the rigid D3h and C2v limits is illustrative, and provides insight into just how hard it is to treat the degenerate bending coordinate (q4) of NO3 accurately.
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Mukherjee, B.; Mukherjee, S.; Sardar, S.; Shamasundar, K. R.; Adhikari, S. An Ab Initio Investigation of Non-Adiabatic Couplings and Conical Intersections Among the Lowest Five Electronic States of the NO₃ Radical. Mol. Phys. 2017, 115, 2833– 2848, DOI: 10.1080/00268976.2017.1340680
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An ab initio investigation of non-adiabatic couplings and conical intersections among the lowest five electronic states of the NO3 radical
Mukherjee, Bijit; Mukherjee, Saikat; Sardar, Subhankar; Shamasundar, K. R.; Adhikari, Satrajit
Molecular Physics (2017), 115 (21-22), 2833-2848CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
In order to explore spectroscopic properties of a floppy mol., it is necessary to compute accurate diabatic potential energy surfaces (PESs) around the Franck-Condon region of nuclear configuration space. In such cases, it is desirable to include the non-adiabatic coupling terms (NACTs) through Jahn-Teller (JT) and pseudo Jahn-Teller (PJT) effects as well as other accidental conical intersections (CIs) prevailing among the electronic states of the mol. In this work, we investigate NACTs among the five lowest doublet electronic states of NO3 radical [ X2A'2(12B2), A2E''(12A2 and 12B1), B2E'(12A1 and 22B2)], in which all those effects (JT, PJT and CIs) have prominent contributions. The study is carried out in terms of normal modes (Qis) of NO3 and the CIs (JT and accidental) are located in a plane of nuclear configuration space of two such coordinates (Qi and Qj). We have calcd. ab initio adiabatic PESs and NACTs of those five states of NO3 at the MRCI level as functions of pairwise Qis. Further, to explore the nature of a given CI in terms of normal coordinates contributing to NACTs within a particular plane Qi-Qj, we have performed CP-MCSCF study computing analytic NACTs and characterised their contributions along various Qks.
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Mukherjee, B.; Mukherjee, S.; Sardar, S.; Shamasundar, K. R.; Adhikari, S. A Beyond Born-Oppenheimer Treatment of Five State Molecular System NO₃ and the Photodetachment Spectra of its Anion. Chem. Phys. 2018 (manuscript in press). DOI: 10.1016/j.chemphys.2018.09.017
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Köppel, H.; Cederbaum, L. S.; Domcke, W. Interplay of Jahn-Teller and pseudo-Jahn-Teller vibronic dynamics in the benzene cation. J. Chem. Phys. 1988, 89, 2023– 2040, DOI: 10.1063/1.455100
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Interplay of Jahn-Teller and pseudo-Jahn-Teller vibronic dynamics in the benzene cation
Koppel, H.; Cederbaum, L. S.; Domcke, W.
Journal of Chemical Physics (1988), 89 (4), 2023-40CODEN: JCPSA6; ISSN:0021-9606.
The static and dynamic aspects of the vibronic interaction of the ~B2E2g and ~C2A2u electronic states of PhH+ were analyzed. In the approxn. of linear vibrational and vibronic coupling, the model Hamiltonian for this system comprises 8 nonseparable vibrational modes, 6 of which are degenerate (2 of A1g symmetry, 4 of E2g symmetry, and 2 of E2u symmetry). The coupling consts. are estd. from existing ab initio SCF and semiempirical (CNDO/S) calcns. The topol. of the adiabatic potential-energy surfaces of this class of model Hamiltonians was investigated. The model exhibits a variety of conical intersections which dominate the vibronic dynamics. The dynamic problem was solved with simultaneous inclusion of 6 vibrational modes, 4 of which are degenerate (the Jahn-Teller coupling of 2 of the E2g modes is negligible). Hamiltonian matrixes with dimensions up to 6 × 106 are diagonalized using the Lanczos algorithm. After some adjustments of coupling consts., the calcn. reproduced the complex structure of the overlapping ~B2E2g-~C2A2u bands in the UPS of PhH. The vibronic structure of the lower-energy E2g band is dominated by a 2-mode Jahn-Teller effect in the ~B state. At higher energy, the marked diffuseness of the ~C2A2u band is a consequence of complete vibronic mixing with the lower-lying ~B state.
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Köppel, H. New Ultrafast Nonradiative Decay Mechanism in the Benzene Radical Cation. Chem. Phys. Lett. 1993, 205, 361– 370, DOI: 10.1016/0009-2614(93)87135-p
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Döscher, M.; Köppel, H. Multiple surface intersections and strong nonadiabatic coupling effects between the D̃²E_1u and Ẽ²B_2u states of C₆H₆⁺. Chem. Phys. 1997, 225, 93– 105, DOI: 10.1016/s0301-0104(97)00228-0
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Döscher, M.; Köppel, H.; Szalay, P. G. Multistate Vibronic Interactions in the Benzene Radical Cation. I. Electronic Structure Calculations. J. Chem. Phys. 2002, 117, 2645– 2656, DOI: 10.1063/1.1491397
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Multistate vibronic interactions in the benzene radical cation. I. Electronic structure calculations
Doscher, Martina; Koppel, Horst; Szalay, Peter G.
Journal of Chemical Physics (2002), 117 (6), 2645-2656CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The multistate vibronic interactions in the benzene radical cation are investigated theor., employing the framework of a linear vibronic coupling scheme. The five lowest electronic states are included in the treatment; in view of the degeneracy of some states, this amts. to eight coupled potential energy surfaces. Different types of ab initio calcns. have been performed for the system parameters and been found to be in good mutual agreement, thus supporting each other. The calcns. reveal a whole sequence of low-energy conical intersections between the potential energy surfaces of different states. Their importance for the nuclear dynamics in this prototypical org. radical cation is pointed out. Wave-packet dynamical simulations for these coupled potential energy surfaces will be presented in the following paper (Paper II).
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Köppel, H.; Döscher, M.; Bâldea, I.; Meyer, H.-D.; Szalay, P. G. Multistate Vibronic Interactions in the Benzene Radical Cation. II. Quantum Dynamical Simulations. J. Chem. Phys. 2002, 117, 2657– 2671, DOI: 10.1063/1.1491398
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Multistate vibronic interactions in the benzene radical cation. II. Quantum dynamical simulations
Koppel, H.; Doscher, M.; Baldea, I.; Meyer, H.-D.; Szalay, P. G.
Journal of Chemical Physics (2002), 117 (6), 2657-2671CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The multistate vibronic dynamics in the ~X 2E1g-~E 2B2u electronic states of the benzene radical cation was studied theor. by an ab initio quantum-dynamical approach. The vibronic coupling scheme and the ab initio values of the system parameters are adopted from the previous Paper I. Vibronic line spectra were obtained with the Lanczos procedure. Extensive calcns. on wave-packet propagation were performed with the aid of the multiconfiguration time-dependent Hartree method. Up to five coupled electronic potential energy surfaces and 13 vibrational degrees of freedom were included in these calcns. As a result, the impact of a 3rd electronic state (~X or ~B) on a strongly coupled manifold (~B-~C or D~-E~ states) is quant. assessed. It leads to a restructuring of the spectral envelope which is stronger for the ~B-D~-E~ than for the X~-B~-C~ system. The internal conversion dynamics was characterized by a stepwise transfer of electronic population to the lowest electronic state on a time scale of ∼100 fs, if the system was prepd. initially on the highest potential energy surface. Companion calcns. also were performed for the case when the system was prepd. in the intermediate state at t = 0; they show a branching of the electronic populations. These are all novel findings which are discussed in terms of conical intersections between the various potential energy surfaces. The importance of such multistate vibronic interactions for the photophysics and photochem. of medium-sized systems is pointed out.
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Köppel, H.; Bâldea, I.; Szalay, P. G. Combined Jahn-Teller and Pseudo-Jahn-Teller Effects in the Benzene Radical Cation. Advances in Quantum Chemistry; Academic Press, 2003; Vol. 44, pp 199– 217.
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Sardar, S.; Paul, A. K.; Sharma, R.; Adhikari, S. A “Classical Trajectory Driven Nuclear Dynamics by a Parallelized Quantum-Classical Approach to a Realistic Model Hamiltonian of Benzene Radical Cation. Int. J. Quantum Chem. 2009, 111, 2741– 2759, DOI: 10.1002/qua.22578
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Sardar, S.; Paul, A. K.; Sharma, R.; Adhikari, S. The Multistate Multimode Vibronic Dynamics of Benzene Radical Cation with a Realistic Model Hamiltonian Using a Parallelized Algorithm of the Quantumclassical Approach. J. Chem. Phys. 2009, 130, 144302, DOI: 10.1063/1.3108488
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The multistate multimode vibronic dynamics of benzene radical cation with a realistic model Hamiltonian using a parallelized algorithm of the quantumclassical approach
Sardar, Subhankar; Paul, Amit Kumar; Sharma, Rahul; Adhikari, Satrajit
Journal of Chemical Physics (2009), 130 (14), 144302/1-144302/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We demonstrate the workability of a parallelized algorithm of the time-dependent discrete variable representation (TDDVR) method to explore the detailed dynamical aspects of vibronic interaction in two three-state model Hamiltonians (X 2E1g, B 2E2g, C 2A2u and B 2E2g, D 2E1u, E 2B2u) of benzene radical cation along with a preliminary investigation on its five electronic states (X 2E1g, B 2E2g, C 2A2u, D 2E1u, and E2B2u). Since those electronic states are interconnected through a series of conical intersections, we have used six and nine vibronically important modes for the three- and five-state Hamiltonians, resp., in order to perform the quantum dynamics on such system. The population profiles calcd. by using our TDDVR approach show reasonably good agreement with the results obtained by exact quantum mech. (multiconfiguration time-dependent Hartree) method, whereas the corresponding (calcd.) photoabsorption spectra originating from various electronic states agree well with the exptl. ones. It is important to note that the parallelized algorithm of our TDDVR approach reduces the computation cost by more than an order of magnitude compared to its serial analog. The TDDVR approach appears to be a good compromise between accuracy and speed for such large mol. system, where quantum mech. description is needed in a restricted region. (c) 2009 American Institute of Physics.
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Sardar, S.; Adhikari, S. A quantum-classical simulation of a multi-surface multi-mode nuclear dynamics on C₆H₆⁺ incorporating degeneracy among electronic states. J. Chem. Sci. 2012, 124, 51– 58, DOI: 10.1007/s12039-011-0195-z
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A quantum-classical simulation of a multi-surface multi-mode nuclear dynamics on incorporating degeneracy among electronic states
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Journal of Chemical Sciences (Bangalore, India) (2012), 124 (1), 51-58CODEN: JCSBB5; ISSN:0974-3626. (Indian Academy of Sciences)
We have performed a nuclear dynamics simulation to calc. photoelectron spectra and population profiles of benzene radical cation (C6H) employing the parallelized Time Dependent Discrete Variable Representation (TDDVR) approach. For this purpose, we have considered two multi-state multi-mode model Hamiltonians of C6H with degeneracy among the eectronic states: (a) One consists of three states and eight modes, which in turn leads to a five state thirteen mode Hamiltonian (X 2E1g -B 2E2g -C 2A2u ) due to the degeneracy; and (b) The other is constituted of three states and thirteen modes which is basically a five state twenty mode Hamiltonian (B 2E2g -D 2E1u -E 2B2u ) for the same reason. Since these electronic states are interconnected by several conical intersections in the vicinity of the Franck Condon region, it will be challenging to pursue such large dynamical calcn. in the presence of nonadiabaticity among the electronic states. The spectral as well as population profiles calcd. with the advent of TDDVR approach show reasonably good agreement with the results obtained by the Multi Configuration Time Dependent Hartree (MCTDH) methodol.
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Goode, J. G.; Hofstein, J. D.; Johnson, P. M. The observation of strong pseudo-Jahn-Teller activity in the benzene cation B̃²E_2g state. J. Chem. Phys. 1997, 107, 1703– 1716, DOI: 10.1063/1.474526
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Johnson, P. M. The Jahn-Teller effect in the lower electronic states of benzene cation. II. Vibrational analysis and coupling constants of the B̃²E_2g state. J. Chem. Phys. 2002, 117, 10001– 10007, DOI: 10.1063/1.1519007
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Johnson, P. M. The Jahn-Teller Effect in the Lower Electronic States of Benzene Cation. I. Calculation of Linear Parameters for the e_2g Modes. J. Chem. Phys. 2002, 117, 9991– 10000, DOI: 10.1063/1.1519006
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The Jahn-Teller effect in the lower electronic states of benzene cation. I. Calculation of linear parameters for the e2g modes
Johnson, Philip M.
Journal of Chemical Physics (2002), 117 (22), 9991-10000CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Jahn-Teller (JT) coupling parameters can be simply derived from features of the potential energy surfaces of the JT active vibrational modes of a mol. subject to that effect. Potential energy curves representing cuts of D2h symmetry have been calcd. using d. functional theory for each of the four active e2g modes of benzene cation in each of the lowest three degenerate electronic states. From these curves and the vibrational frequencies, the JT coupling parameters can be found (vibrational numbering follows Wilson's notation, coupling parameters are in Moffit's notation). For the C6H+6X 2E1g state the largest JT coupling parameter is provided by mode 6 (D=0.48), followed closely by mode 8 (0.30) and then mode 9 (0.10). Mode 7 provides almost no stabilization. For the B 2E2g state, calcd. coupling parameters are very large, particularly for modes 8 (1.36) and 6 (0.93). Modes 7 (0.10) and 9 (0.07) are smaller but finite. For the D 2E1u state, mode 6 has an impressive linear coupling parameter of 4.12, able to support several vibrations below the JT cusp. Indications of a substantial quadratic stabilization for that mode result in the possibility of a completely static distortion in that state. Similar, but not identical, results are obtained for C6D6+.
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Ghosh, S.; Mukherjee, S.; Mukherjee, B.; Mandal, S.; Sharma, R.; Chaudhury, P.; Adhikari, S. Beyond Born-Oppenheimer theory for ab initio constructed diabatic potential energy surfaces of singlet H₃⁺ to study reaction dynamics using coupled 3D time-dependent wave-packet approach. J. Chem. Phys. 2017, 147, 074105, DOI: 10.1063/1.4998406
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Beyond Born-Oppenheimer theory for ab initio constructed diabatic potential energy surfaces of singlet H3+ to study reaction dynamics using coupled 3D time-dependent wave-packet approach
Ghosh, Sandip; Mukherjee, Saikat; Mukherjee, Bijit; Mandal, Souvik; Sharma, Rahul; Chaudhury, Pinaki; Adhikari, Satrajit
Journal of Chemical Physics (2017), 147 (7), 074105/1-074105/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The workability of beyond Born-Oppenheimer theory to construct diabatic potential energy surfaces (PESs) of a charge transfer atom-diatom collision process has been explored by performing scattering calcns. to ext. accurate integral cross sections (ICSs) and rate consts. for comparison with most recent exptl. quantities. We calc. non-adiabatic coupling terms among the lowest three singlet states of H3 + system (11A', 21A', and 31A') using MRCI level of calcn. and solve the adiabatic-diabatic transformation equation to formulate the diabatic Hamiltonian matrix of the same process [S. Mukherjee et al., J. Chem. Phys. 141, 204306 (2014)] for the entire region of nuclear configuration space. The nonadiabatic effects in the D+ + H2 reaction has been studied by implementing the coupled 3D time-dependent wave packet formalism in hyperspherical coordinates [S. Adhikari and A. J. C. Varandas, Comput. Phys. Commun. 184, 270 (2013)] with zero and non-zero total angular momentum (J) on such newly constructed accurate (ab initio) diabatic PESs of H3+. We have depicted the convergence profiles of reaction probabilities for the reactive non-charge transfer, non-reactive charge transfer, and reactive charge transfer processes for different collisional energies with respect to the helicity (K) and total angular momentum (J) quantum nos. Finally, total and state-to-state ICSs are calcd. as a function of collision energy for the initial rovibrational state (v = 0, j = 0) of the H2 mol., and consequently, those quantities are compared with previous theor. and exptl. results. (c) 2017 American Institute of Physics.
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Koushik Naskar, Soumya Mukherjee, Bijit Mukherjee, Satyam Ravi, Saikat Mukherjee, Subhankar Sardar, Satrajit Adhikari. ADT: A Generalized Algorithm and Program for Beyond Born–Oppenheimer Equations of “N” Dimensional Sub-Hilbert Space. Journal of Chemical Theory and Computation 2020, 16 (3) , 1666-1680. https://doi.org/10.1021/acs.jctc.9b00948
Soumya Mukherjee, Koushik Naskar, Saikat Hazra, Mantu Kumar Sah, Satrajit Adhikari. Beyond Born-Oppenheimer Treatment for Multi-State Photoelectron Spectra, Phase Transitions of Solids and Scattering Processes. Journal of Physics: Conference Series 2024, 2769 (1) , 012012. https://doi.org/10.1088/1742-6596/2769/1/012012
Mantu Kumar Sah, Soumya Mukherjee, Swagato Saha, Koushik Naskar, Satrajit Adhikari. Photoelectron spectra of benzene: Can path dependent diabatic surfaces provide unique observables?. The Journal of Chemical Physics 2023, 159 (24) https://doi.org/10.1063/5.0177186
Saikat Hazra, Soumya Mukherjee, Satyam Ravi, Subhankar Sardar, Satrajit Adhikari. Construction of Beyond Born‐Oppenheimer Based Diabatic Surfaces and Generation of Photoabsorption Spectra: The Touchstone Pyrazine (C 4 N 2 H 4 ). ChemPhysChem 2022, 23 (23) https://doi.org/10.1002/cphc.202200482
Soumya Mukherjee, Satyam Ravi, Joy Dutta, Subhankar Sardar, Satrajit Adhikari. Beyond Born–Oppenheimer based diabatic surfaces of 1,3,5-C 6 H 3 F 3 + to generate the photoelectron spectra using time-dependent discrete variable representation approach. Physical Chemistry Chemical Physics 2022, 24 (4) , 2185-2202. https://doi.org/10.1039/D1CP04733G
Soumya Mukherjee, Satyam Ravi, Koushik Naskar, Subhankar Sardar, Satrajit Adhikari. A beyond Born–Oppenheimer treatment of C6H6+ radical cation for diabatic surfaces: Photoelectron spectra of its neutral analog using time-dependent discrete variable representation. The Journal of Chemical Physics 2021, 154 (9) https://doi.org/10.1063/5.0040361
Joy Dutta, Soumya Mukherjee, Koushik Naskar, Sandip Ghosh, Bijit Mukherjee, Satyam Ravi, Satrajit Adhikari. The role of electron–nuclear coupling on multi-state photoelectron spectra, scattering processes and phase transitions. Physical Chemistry Chemical Physics 2020, 22 (47) , 27496-27524. https://doi.org/10.1039/D0CP04052E
Bijit Mukherjee, Koushik Naskar, Soumya Mukherjee, Sandip Ghosh, Tapas Sahoo, Satrajit Adhikari. Beyond Born–Oppenheimer theory for spectroscopic and scattering processes. International Reviews in Physical Chemistry 2019, 38 (3-4) , 287-341. https://doi.org/10.1080/0144235X.2019.1672987
Soumya Mukherjee, Joy Dutta, Bijit Mukherjee, Subhankar Sardar, Satrajit Adhikari. Conical intersections and nonadiabatic coupling terms in 1,3,5-C6H3F3+: A six state beyond Born-Oppenheimer treatment. The Journal of Chemical Physics 2019, 150 (6) https://doi.org/10.1063/1.5064519
Bijit Mukherjee, Sandip Ghosh, Satrajit Adhikari. Beyond Born-Oppenheimer treatment on spectroscopic and scattering processes. Journal of Physics: Conference Series 2018, 1148 , 012001. https://doi.org/10.1088/1742-6596/1148/1/012001

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Abstract
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Figure 1
Figure 1. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–5) are depicted for (a) ground, (b) first excited, (c) second excited, (d) third excited, and (e) fourth excited states of NO₃ radical along Q_3x normal mode taking average over Q_3y. All of the PECs exhibit symmetric variation and nonlinear functional characteristics.
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Figure 2
Figure 2. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–5) are represented for (a) ground, (b) first excited, (c) second excited, (d) third excited, and (e) fourth excited electronic states of NO₃ along Q_3y normal mode taking average over Q_3x. Here, the PECs are asymmetric and incorporate linear functionalities.
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Figure 3
Figure 3. Dressed diabatic coupling terms of NO₃ radical along Q_3x normal mode: (a) and (b) are smooth, symmetric as well as continuous appeared due to “2–3” and “4–5” JT CIs and three “4–5” accidental CIs. On the other hand, symmetric and single-valued (c) and (d) represent PJT couplings. Therefore, JT activity is originated solely because of nonlinear couplings along the Q_3x mode.
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Figure 4
Figure 4. Dressed diabatic coupling terms of NO₃ radical along the Q_3y normal mode. Here, (a) , (b) , (c) , and (d) all are smooth and continuous but the variation is asymmetric. Along Q_3y normal mode, JT coupling contains linear as well as nonlinear functionalities.
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Figure 5
Figure 5. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–4) are plotted for low-lying four electronic states of Bz⁺ radical cation in diagrams (a–d), respectively, along Q_16x normal mode taking average over the other component, Q_16y. All PECs undergo symmetric variation and include only the even order functional forms.
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Figure 6
Figure 6. Dressed adiabatic (), diabatic (), and adiabatic-via-dressed diabatic () PECs (n = 1–4) are portrayed for lowest four electronic states of Bz⁺ radical cation in diagrams (a–d), respectively, along the Q_16y normal mode by averaging over Q_16x. Here, PECs are asymmetric and incorporate both linear and nonlinear terms.
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Figure 7
Figure 7. Dressed diabatic coupling terms are plotted along Q_16x normal mode of the Bz⁺ radical cation. Figures (a,b) depict and , respectively. Both are symmetric, indicating nonlinear “1–2” and “3–4” JT interactions.
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Figure 8
Figure 8. Dressed diabatic coupling terms are plotted with the variation of Q_16y normal mode coordinates of the Bz⁺ radical cation. Figures (a,b) describe and , respectively. Along Q_16y normal mode, JT coupling contains linear functions along with nonlinear terms.
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  There is no corresponding record for this reference.
11. 11
  Feynman, R. P. Forces in Molecules. Phys. Rev. 1939, 56, 340– 343, DOI: 10.1103/physrev.56.340
  11
  Forces in molecules
  Feynman, R. P.
  Physical Review (1939), 56 (), 340-3CODEN: PHRVAO; ISSN:0031-899X.
  Wave-mech. formulas are developed to calc. the forces in a mol. system directly, rather than indirectly through the agency of energy.
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12. 12
  Baer, M. Introduction to the Theory of Electronic Non-Adiabatic Coupling Terms in Molecular Systems. Phys. Rep. 2002, 358, 75– 142, DOI: 10.1016/s0370-1573(01)00052-7
  12
  Introduction to the theory of electronic non-adiabatic coupling terms in molecular systems
  Baer, Michael
  Physics Reports (2002), 358 (2), 75-142CODEN: PRPLCM; ISSN:0370-1573. (Elsevier Science B.V.)
  A review. The Born-Oppenheimer treatment leads to the adiabatic framework where the nonadiabatic terms are the phys. entities responsible for the coupling between adiabatic states. The main disadvantage of this treatment is in the fact that these coupling terms frequently become singular thus causing difficulties in solving the relevant Schrodinger equation for the motion of the nuclei that make up the mol. systems. In this review, we present the line integral approach which enables the formation of the adiabatic-to-diabatic transformation matrix that yields the friendlier diabatic framework. The review concs. on the math. conditions that allow the rigorous derivation of the adiabatic-to-diabatic transformation matrix and its interesting phys. properties. One of the findings of this study is that the nonadiabatic coupling terms have to be quantized in a certain manner in order to yield single-valued diabatic potentials. Another important feature revealed is the existence of the topol. matrix, which contains all the topol. features of a given mol. system related to a closed contour in configuration space. Finally, we present an approxn. that results from the Born-Oppenheimer treatment which, in contrast to the original Born-Oppenheimer approxn., contains the effect of the nonadiabatic coupling terms. The various derivations are accompanied by examples which in many cases are interesting by themselves.
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13. 13
  Baer, M. Beyond Born–Oppenheimer: Conical Intersections and Electronic Nonadiabatic Coupling Terms; Wiley Interscience: NJ, 2006.
  There is no corresponding record for this reference.
14. 14
  Baer, M.; Englman, R. A Study of the Diabatic Electronic Representation within the Born-Oppenheimer Approximation. Mol. Phys. 1992, 75, 293– 303, DOI: 10.1080/00268979200100231
  14
  A study of the diabatic electronic representation within the Born-Oppenheimer approximation
  Baer, Michael; Englman, Robert
  Molecular Physics (1992), 75 (2), 293-303CODEN: MOPHAM; ISSN:0026-8976.
  The adiabatic-diabatic transformation is considered 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 sepd. everywhere in configuration space and the other is the case of conical intersections exemplified by the E ⊗ ε Jahn-Teller situation. In both cases well defined diabatic states can be formed in most regions of configuration space.
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15. 15
  Mebel, A. M.; Halász, G. J.; Vibók, Á.; Alijah, A.; Baer, M. Quantization of the 3×3 nonadiabatic coupling matrix for three coupled states of the C₂H molecule. J. Chem. Phys. 2002, 117, 991– 1000, DOI: 10.1063/1.1483854
  There is no corresponding record for this reference.
16. 16
  Sarkar, B.; Adhikari, S. Extended Born-Oppenheimer Equation for a Three-State System. J. Chem. Phys. 2006, 124, 074101, DOI: 10.1063/1.2170089
  16
  Extended Born-Oppenheimer equation for a three-state system
  Sarkar, Biplab; Adhikari, Satrajit
  Journal of Chemical Physics (2006), 124 (7), 074101/1-074101/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrixes are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity anal. around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calcns. are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.
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17. 17
  Sarkar, B.; Adhikari, S. Curl Condition for a Four-State Born–Oppenheimer System Employing the Mathieu Equation. J. Phys. Chem. A 2008, 112, 9868– 9885, DOI: 10.1021/jp8029709
  17
  Curl Condition for a Four-State Born-Oppenheimer System Employing the Mathieu Equation
  Sarkar, Biplab; Adhikari, Satrajit
  Journal of Physical Chemistry A (2008), 112 (40), 9868-9885CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  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 Schrodinger equation (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 anal. validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling elements, namely, the validity of curl conditions and the nature of curl equations around conical intersections.
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18. 18
  Paul, A. K.; Sardar, S.; Sarkar, B.; Adhikari, S. Single Surface Beyond Born-Oppenheimer Equation for a Three-State Model Hamiltonian of Na₃ Cluster. J. Chem. Phys. 2009, 131, 124312, DOI: 10.1063/1.3236839
  There is no corresponding record for this reference.
19. 19
  Mukherjee, S.; Mukherjee, B.; Adhikari, S. Five Electronic State Beyond Born-Oppenheimer Equations and Their Applications to Nitrate and Benzene Radical Cation. J. Phys. Chem. A 2017, 121, 6314– 6326, DOI: 10.1021/acs.jpca.7b04592
  19
  Five Electronic State Beyond Born-Oppenheimer Equations and Their Applications to Nitrate and Benzene Radical Cation
  Mukherjee, Soumya; Mukherjee, Bijit; Adhikari, Satrajit
  Journal of Physical Chemistry A (2017), 121 (33), 6314-6326CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  We present explicit form of Adiabatic to Diabatic Transformation (ADT) equations and expressions of nonadiabatic coupling terms (NACTs) for a coupled five-state electronic manifold in terms of ADT angles between electronic wavefunctions. ADT matrixes eliminate the numerical instability arising from singularity of NACTs and transform the adiabatic Schrodinger equation to its diabatic form. Two real mol. systems NO3 and C6H6+ (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 [X2A2' (12B2), A2E''(12A2 and 12B1) and B2E'(12A1 and 22B2)], in which all types of non-adiabatic interactions i.e. 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+ [X2E1g (12B3g and 12B2g), B2E2g (12Ag and 12B1g) and C2A2u (12B1u)] depict similar kind of complex feature of non-adiabatic effects. For NO3 radical, the two components of degenerate in-plane asym. 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 asym. stretching mode and the other one is constituted with one of the components each from out-of-plane degenerate bend and in-plane degenerate asym. stretching modes. We calc. ab initio adiabatic potential energy surfaces (PESs) and NACTs among the lowest five electronic states at the CASSCF level using MOLPRO quantum chem. package. Subsequently, the ADT is carried out using those newly developed equations to validate the positions of the CIs, evaluate the ADT angles and construct smooth, sym. and continuous diabatic PESs for both the mol. systems.
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20. 20
  Paul, A. K.; Ray, S.; Mukhopadhyay, D.; Adhikari, S. Ab initio Calculations on the Excited States of Na₃ Cluster to Explore Beyond Born-Oppenheimer Theories: Adiabatic to Diabatic Potential Energy Surfaces and Nuclear Dynamics. J. Chem. Phys. 2011, 135, 034107, DOI: 10.1063/1.3609247
  20
  Ab initio calculations on the excited states of Na3 cluster to explore beyond Born-Oppenheimer theories: Adiabatic to diabatic potential energy surfaces and nuclear dynamics
  Paul, Amit Kumar; Ray, Somrita; Mukhopadhyay, Debasis; Adhikari, Satrajit
  Journal of Chemical Physics (2011), 135 (3), 034107/1-034107/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We perform ab initio calcn. using quantum chem. package (MOLPRO) on the excited states of Na3 cluster and present the adiabatic PESs for the electronic states 22E' and 12A'1, and the non-adiabatic coupling (NAC) terms among those states. Since the ab initio calcd. NAC elements for the states 22E' and 12A'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 theor. means on beyond Born-Oppenheimer treatment. Nuclear dynamics has been carried out on those diabatic surfaces to reproduce the exptl. spectrum for system B of Na3 cluster and thereby, to explore the numerical validity of the theor. development on beyond Born-Oppenheimer approach for adiabatic to diabatic transformation. (c) 2011 American Institute of Physics.
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21. 21
  Mukherjee, S.; Bandyopadhyay, S.; Paul, A. K.; Adhikari, S. Construction of Diabatic Hamiltonian Matrix from Ab Initio Calculated Molecular Symmetry Adapted Nonadiabatic Coupling Terms and Nuclear Dynamics for the Excited States of Na₃ Cluster. J. Phys. Chem. A 2013, 117, 3475– 3495, DOI: 10.1021/jp311597c
  21
  Construction of Diabatic Hamiltonian Matrix from ab Initio Calculated Molecular Symmetry Adapted Nonadiabatic Coupling Terms and Nuclear Dynamics for the Excited States of Na3 Cluster
  Mukherjee, Saikat; Bandyopadhyay, Sudip; Paul, Amit Kumar; Adhikari, Satrajit
  Journal of Physical Chemistry A (2013), 117 (16), 3475-3495CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  We present the mol. symmetry (MS) adapted treatment of nonadiabatic coupling terms (NACTs) for the excited electronic states (22E' and 12A1') of Na3 cluster, where the adiabatic potential energy surfaces (PESs) and the NACTs are calcd. at the MRCI level by using an ab initio quantum chem. package (MOLPRO). The signs of the NACTs at each point of the configuration space (CS) are detd. 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 cor. NACTs and the corresponding ADT angles demonstrate the validity of curl condition for the existence of three-state (22E' and 12A1') sub-Hilbert space, it becomes possible to construct the continuous, single-valued, sym., 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 the exptl. spectrum for system B of Na3 cluster.
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22. 22
  Mukherjee, S.; Adhikari, S. The Excited States of K₃ Cluster: The Molecular Symmetry Adapted Non-Adiabatic Coupling Terms and Diabatic Hamiltonian Matrix. Chem. Phys. 2014, 440, 106– 118, DOI: 10.1016/j.chemphys.2014.05.022
  22
  The excited states of K3 cluster: The molecular symmetry adapted non-adiabatic coupling terms and diabatic Hamiltonian matrix
  Mukherjee, Saikat; Adhikari, Satrajit
  Chemical Physics (2014), 440 (), 106-118CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
  We calc. the adiabatic potential energy surfaces (PESs) and the nonadiabatic coupling terms (NACTs) for the excited electronic states of K3 cluster by MRCI approach using MOLPRO. The NACTs are adapted with mol. symmetry to assign appropriate IREPs so that the elements of the Hamiltonian matrix are totally sym. We incorporate those NACTs into three-state adiabatic-to-diabatic transformation (ADT) equations to obtain ADT angles for constructing continuous, single-valued, smooth and sym. diabatic Hamiltonian matrix, where its elements are fitted with analytic functions. Finally, we demonstrate that the dressed diabatic and adiabatic-via-dressed diabatic PECs show prominent topol. effect over dressed adiabatic curves.
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23. 23
  Mukherjee, S.; Mukhopadhyay, D.; Adhikari, S. Conical Intersections and Diabatic Potential Energy Surfaces for the Three Lowest Electronic Singlet States of H₃⁺. J. Chem. Phys. 2014, 141, 204306, DOI: 10.1063/1.4901986
  23
  Conical intersections and diabatic potential energy surfaces for the three lowest electronic singlet states of H3+
  Mukherjee, Saikat; Mukhopadhyay, Debasis; Adhikari, Satrajit
  Journal of Chemical Physics (2014), 141 (20), 204306/1-204306/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We calc. the adiabatic Potential Energy Surfaces (PESs) and the Non-Adiabatic Coupling Terms (NACTs) for the three lowest singlet states of H3+ in hyperspherical coordinates as functions of hyperangles (θ and .vphi.) for a grid of fixed values of hyperradius (1.5 ≤ ρ ≤ 20 bohrs) using the MRCI level of methodol. employing ab initio quantum chem. package (MOLPRO). The NACT between the ground and the first excited state translates along the seams on the θ - .vphi. space, i.e., there are six Conical Intersections (CIs) at each θ (60° ≤ θ ≤ 90°) within the domain, 0 ≤ .vphi. ≤ 2π. While transforming the adiabatic PESs to the diabatic ones, such surfaces show up six crossings along those seams. Our beyond Born-Oppenheimer approach could incorporate the effect of NACTs accurately and construct single-valued, continuous, smooth, and sym. diabatic PESs. Since the location of CIs and the spatial amplitudes of NACTs are most prominent around ρ = 10 bohrs, generally only those results are depicted. (c) 2014 American Institute of Physics.
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24. 24
  Mukherjee, S.; Mukherjee, B.; Sardar, S.; Adhikari, S. Ab Initio Constructed Diabatic Surfaces of NO₂ and the Photodetachment Spectra of its Anion. J. Chem. Phys. 2015, 143, 244307, DOI: 10.1063/1.4938526
  24
  Ab initio constructed diabatic surfaces of NO2 and the photodetachment spectra of its anion
  Mukherjee, Saikat; Mukherjee, Bijit; Sardar, Subhankar; Adhikari, Satrajit
  Journal of Chemical Physics (2015), 143 (24), 244307/1-244307/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A thorough investigation has been performed for electronic structure, topol. effect, and nuclear dynamics of NO2 mol., where the adiabatic potential energy surfaces (PESs), conical intersections between the ground (X2A1) and the first excited state (A2B2), and the corresponding non-adiabatic coupling terms between those states are recalcd. [Chem. Phys. 416, 11 (2013)] to achieve enough accuracy in dynamics. We employ beyond Born-Oppenheimer theory for these two state sub-Hilbert space to carry out adiabatic to diabatic transformation (ADT) to obtain the ADT angles and thereby, to construct single-valued, smooth, and continuous diabatic PESs. The analytic expressions for the adiabatic PESs and ADT angles are provided to represent a two-state three-mode diabatic Hamiltonian of NO2 for performing nuclear dynamics to calc. the photo-electron spectra of its anion. It appears that not only Jahn-Teller type coupling but also Renner-Teller interaction contributes significantly on the overall spectrum. The coupling between the electronic states (X2A1 and A2B2) of NO2 is essentially through the asym. stretching mode, where the functional form of such interaction is distinctly sym. and non-linear. (c) 2015 American Institute of Physics.
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25. 25
  Nelson, H. H.; Pasternack, L.; McDonald, J. R. Laser-Induced Excitation and Emission Spectra of Nitrate Radical (NO₃). J. Phys. Chem. 1983, 87, 1286– 1288, DOI: 10.1021/j100231a003
  25
  Laser-induced excitation and emission spectra of nitrate radical (NO3)
  Nelson, H. H.; Pasternack, Louise; McDonald, J. R.
  Journal of Physical Chemistry (1983), 87 (8), 1286-8CODEN: JPCHAX; ISSN:0022-3654.
  The fluorescence excitation and emission spectra of NO3 (2E' (2B2) ↔ 2A2' (2B2)) are reported. Assignments were made for excited- and ground-state vibrational frequencies. A lower limit of 23 μs was measured for the fluorescence lifetime which is at least a factor of 40 longer than predicted from absorption. Spectroscopic observations are consistent with a C2v ground-state geometry.
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26. 26
  Ishiwata, T.; Fujiwara, I.; Naruge, Y.; Obi, K.; Tanaka, I. Study of Nitrate Radical by Laser-Induced Fluorescence. J. Phys. Chem. 1983, 87, 1349– 1352, DOI: 10.1021/j100231a016
  26
  Study of nitrate radical by laser-induced fluorescence
  Ishiwata, Takashi; Fujiwara, Ichiro; Naruge, Yukio; Obi, Kinichi; Tanaka, Ikuzo
  Journal of Physical Chemistry (1983), 87 (8), 1349-52CODEN: JPCHAX; ISSN:0022-3654.
  The nitrate radical, NO3, was directly detected by the laser-induced fluorescence method. The band structure of the fluorescence excitation spectrum was consistent with that of the absorption spectrum. The fluorescence spectrum excited at 662 nm corresponding to the 0-0 transition showed progressions with 1060- and 1480-cm-1 intervals, which were assigned to the sym. stretching (ν1) and degenerate antisym. stretching (ν3) modes of the ground state NO3, resp. The fluorescence lifetime excited at the 0-0 band was estd. to be 2.8 μs from the Stern-Volmer plots in the pressure region of 0.04-0.6 torr.
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27. 27
  Ishiwata, T.; Tanaka, I.; Kawaguchi, K.; Hirota, E. Infrared diode laser spectroscopy of the NO₃ ν₃ band. J. Chem. Phys. 1985, 82, 2196– 2205, DOI: 10.1063/1.448362
  27
  Infrared diode laser spectroscopy of the nitrate radical (NO3)ν3 band
  Ishiwata, Takashi; Tanaka, Ikuzo; Kawaguchi, Kentarou; Hirota, Eizi
  Journal of Chemical Physics (1985), 82 (5), 2196-205CODEN: JCPSA6; ISSN:0021-9606.
  The N-O degenerate stretching band ν3 of the NO3 radical was studied in the gas phase by IR tunable diode laser spectroscopy. The NO3 radical was generated by the reaction of NO2 with an excess of O3. Zeeman modulation was used to observe the paramagnetic absorption lines of 14NO3 and 15NO3 in the wavelength regions 1480-1500 and 1463-1479 cm-1, resp. Only K'' = 3n(n denoting an integer) transitions were obsd., and the N'' = even members were missing from the K'' = 0 manifold. These observations indicate that the NO3 radical belongs to D3h symmetry in the 2A2' ground electronic state. The obsd. spectrum was analyzed by using a sym.-top vibration-rotation Hamiltonian including the spin-rotation interaction. The main parameters thus obtained for 14NO3 are B3 = 0.455 22(11), C3 = 0.227 13(6), Cζ3 = 0.044 79(11), q3 = 0.001 624(33), t3 = 0.000 000 458 0(43), B0 = 0.457 46(12), C0 = B0/2 (fixed), εbb = 0.0280(27), and εcc = 0.1197(36) for v3 = 1, εbb = 0.0277(28), and εcc = 0.1117(34) for v = 0, and ν0 = 1492.3929(9), all in cm-1 with 1 std. error in parentheses. Although these parameters well reproduced the obsd. spectrum, the following anomalous features were noted: (1) a large εcc spin-rotation interaction const. was required to explain the spin splittings for both the ν3 and ground states, (2) a higher-order vibration-rotation interaction term having Δk = ± 4 and Δl = .-+. 2 needed to be included, with the corresponding interaction const. t3 larger than that of CHF3, and (3) the centrifugal distortion consts. and the 1st order Coriolis coupling const. which were derived did not agree with those calcd. assuming a reasonable force field.
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28. 28
  Hirota, E.; Ishiwata, T.; Kawaguchi, K.; Fujitake, M.; Ohashi, N.; Tanaka, I. Near-infrared Band of the Nitrate Radical NO₃ Observed by Diode Laser Spectroscopy. J. Chem. Phys. 1997, 107, 2829– 2838, DOI: 10.1063/1.474641
  28
  Near-infrared band of the nitrate radical NO3 observed by diode laser spectroscopy
  Hirota, Eizi; Ishiwata, Takashi; Kawaguchi, Kentarou; Fujitake, Masaharu; Ohashi, Nobukimi; Tanaka, Ikuzo
  Journal of Chemical Physics (1997), 107 (8), 2829-2838CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The near-IR band of NO3 obsd. at 7602 cm-1 was analyzed by using diode laser spectroscopy. Most of the spectral lines were recorded using source-frequency modulation. Zeeman modulation was found useful in selectively detecting some Q branch lines, which provided a clue to the assignment of the obsd. spectra. The band satisfied selection rules for a parallel band and was thus ascribed to a 2A1''-2A2' vibronic component assocd. with the 2E''-~X2A2' electronic transition, namely, to a transition from the ground vibronic state to the A1'' vibronic state resulting from excitation of the degenerate in-plane bending mode in the 2E'' electronically excited state manifold. The band was almost free of perturbations, except for some K = 6 lines. The least-squares anal. of 581 assigned lines led to mol. parameters of the upper state, where ground-state parameters were fixed to those obtained from the IR study previously reported. The upper-state B rotational const. gave the effective N-O distance of 1.271 Å, which is to be compared with 1.240 Å in the ground vibronic state. The εbb spin-rotation interaction const. of the upper state was close in magnitude to that in the ground vibronic state, but of opposite sign. This observation indicates that the spin-rotation interaction is primarily caused by that between the 2E'' excited and the ground electronic states.
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29. 29
  Weaver, A.; Arnold, D. W.; Bradforth, S. E.; Neumark, D. M. Examination of the ²A₂′ and ²E′ states of NO₃ by ultraviolet photoelectron spectroscopy of NO₃^–. J. Chem. Phys. 1991, 94, 1740– 1751, DOI: 10.1063/1.459947
  29
  Examination of the 2A2' and 2E'' states of nitrogen oxide (NO3) by ultraviolet photoelectron spectroscopy of the nitrate ion
  Weaver, A.; Arnold, D. W.; Bradforth, S. E.; Neumark, D. M.
  Journal of Chemical Physics (1991), 94 (3), 1740-51CODEN: JCPSA6; ISSN:0021-9606.
  The photoelectron spectrum of the NO3- anion was obtained at 266 and at 213 nm. The 266 nm spectrum probes the 2A2' ground state of NO3. The 213 nm spectrum represents the first observation of the 2E'' lowest-lying excited state of NO3. The 2A2' band shows vibrational progressions in the ν1 sym. stretch and the ν4 degenerate in-plane bend of NO3. The anal. of this band indicates that the NO3 ground state has a D3h equil. geometry and is vibronically coupled to the 2E' second excited state via the ν4 mode. Also obtained was the electron affinity of NO3, 3.937 ± 0.014 eV, and the heat of formation of NO3 at 298 K, 0.777 ± 0.027 eV (17.9 ± 0.6 kcal/mol). The 2E'' state of NO3 lies 0.868 ± 0.014 eV above the ground state. The 2E'' band shows complex and extensive vibrational structure. Several possible assignments of this structure are discussed.
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30. 30
  Deev, A.; Sommar, J.; Okumura, M. Cavity ringdown spectrum of the forbidden Ã²E″ ← X̃²A₂′ transition of NO₃: Evidence for static Jahn-Teller distortion in the Ã state. J. Chem. Phys. 2005, 122, 224305, DOI: 10.1063/1.1897364
  There is no corresponding record for this reference.
31. 31
  Codd, T.; Chen, M.-W.; Roudjane, M.; Stanton, J. F.; Miller, T. A. Jet cooled cavity ringdown spectroscopy of the Ã²E″ ← X̃²A₂′ transition of the NO₃ radical. J. Chem. Phys. 2015, 142, 184305, DOI: 10.1063/1.4919690
  31
  Jet cooled cavity ringdown spectroscopy of the ~A2E'' ← ~X2A2' transition of the NO3 radical
  Codd, Terrance; Chen, Ming-Wei; Roudjane, Mourad; Stanton, John F.; Miller, Terry A.
  Journal of Chemical Physics (2015), 142 (18), 184305/1-184305/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The ~A2E←~X2A'2 spectrum of NO3 radical from 7550 cm-1 to 9750 cm-1 was recorded and analyzed. The spectrum differs from previously recorded spectra of this transition due to jet-cooling, which narrows the rotational contours and eliminates spectral interference from hot bands. Assignments of numerous vibronic features can be made based on both band contour and position including the previously unassigned 310 band and several assocd. combination bands. The authors analyzed the spectrum 1st with an independent anharmonic oscillator model and then by a quadratic Jahn-Teller vibronic coupling model. The fit achieved with the quadratic Jahn-Teller model is excellent, but the potential energy surface obtained with the fitted parameters is in only qual. agreement with 1 obtained from ab initio calcns. (c) 2015 American Institute of Physics.
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32. 32
  Eisfeld, W.; Morokuma, K. A Detailed Study on the Symmetry Breaking and its Effect on the Potential Surface of NO₃. J. Chem. Phys. 2000, 113, 5587– 5597, DOI: 10.1063/1.1290607
  32
  A detailed study on the symmetry breaking and its effect on the potential surface of NO3
  Eisfeld, Wolfgang; Morokuma, Keiji
  Journal of Chemical Physics (2000), 113 (14), 5587-5597CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The tenacious symmetry breaking of the electronic wave function of the nitrate radical (NO3) and its effect on the ground-state potential energy surface is investigated in detail. The symmetry breaking of Hartree-Fock wave functions results from a dominance of the orbital localization effect over the resonance effect and leads to three different solns., one sym. and two distorted ones, for the same electronic state. The resp. equil. geometries of these solns. are points on different potential surfaces, making their comparison meaningless. The resonance effect is promoted by dynamic as well as static electron correlation. However, the dynamic correlation methods [e.g., many-body perturbation theory (MBPT) and coupled-cluster single double (CCSD)] cannot overcome the symmetry breaking of the ref. function and the problem of multiple solns. persists. The symmetry breaking can be avoided by the complete active space SCF (CASSCF) approach that yields unique, single-valued surfaces for all electronic states. However, a sufficiently large and appropriately selected active space has to be used to avoid unphys. distortion of the wave function. Still the orbital localization effect leads to equil. geometries of C2v symmetry which strongly depend on the state-averaging of the CASSCF wave function. Multireference single double CI (MR-SDCI) wave functions are also free of symmetry breaking, if the ref. orbitals are and if the configuration space is invariant under the symmetry operations. MRCI geometry optimizations only result in D3h sym. structures with bond lengths and harmonic frequencies in close agreement with exptl. data.
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33. 33
  Mayer, M.; Cederbaum, L. S.; Köppel, H. Ground State Dynamics of NO₃: Multimode Vibronic Borrowing Including Thermal Effects. J. Chem. Phys. 1994, 100, 899– 911, DOI: 10.1063/1.466572
  33
  Ground state dynamics of NO3: multimode vibronic borrowing including thermal effects
  Mayer, M.; Cederbaum, L. S.; Koeppel, H.
  Journal of Chemical Physics (1994), 100 (2), 899-911CODEN: JCPSA6; ISSN:0021-9606.
  The X2A2' band of the photoelectron spectrum of NO3- is theor. calcd. in the framework of a multimode vibronic coupling model. Linear coupling between and within the X2A2' and the B2E' electronic states of NO3 is the important mechanism governing the dynamics. The necessary coupling consts. are obtained from the ionization potentials of NO3- calcd. at different geometries using ab initio Green's functions. Comparison of the theor. with exptl. results [A. Weaver, D. W. Arnold, S. E. Bradforth, and D. M. Neumark, J. Chem. Phys. 94(3), 1740(1991)] for the PE spectrum shows good agreement if a temp. of the NO3- anion of 455 K in the expt. is assumed. A general theor. treatment of thermal effects in vibronically coupled electronic states is presented. The ground electronic state potential surface of NO3 is discussed in the framework of the vibronic coupling model. A shallow min. at a C2v geometry with a barrier height of 0.006 eV relative to the min. energy D3h configuration is found. It is too weak to deform the effective geometry of the mol. from D3h to C2v. Mode-mode coupling is found to play a relevant role in the ground electronic state of NO3, in general.
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34. 34
  Faraji, S.; Köppel, H.; Eisfeld, W.; Mahapatra, S. Towards a higher-order description of Jahn-Teller coupling effects in molecular spectroscopy: The state of NO₃. Chem. Phys. 2008, 347, 110– 119, DOI: 10.1016/j.chemphys.2007.10.006
  There is no corresponding record for this reference.
35. 35
  Okumura, M.; Stanton, J. F.; Deev, A.; Sommar, J. New insights into the Jahn-Teller effect in NO₃ via the dark Ã²E″ state. Phys. Scr. 2006, 73, C64– C70, DOI: 10.1088/0031-8949/73/1/n12
  35
  New insights into the Jahn-Teller effect in NO3 via the dark A 2E'' state
  Okumura, Mitchio; Stanton, John F.; Deev, Andrei; Sommar, Jonas
  Physica Scripta (2006), 73 (1), C64-C70CODEN: PHSTBO; ISSN:0031-8949. (Institute of Physics Publishing)
  The recent cavity ringdown (CRD) measurement of the forbidden ∼A 2E'' ← X 2A2' transition of the nitrate radical NO3 reveals a rich, well-resolved spectrum in the near-IR. The spectroscopic detail provides a new window onto the Jahn-Teller (JT) and pseudo-Jahn-Teller (PJT) effects in NO3. This paper reviews the current exptl. evidence for vibronic coupling in the ∼A state and discusses the theor. issues in the context of new preliminary EOMIP/CCSD and CCSD(T) calcns. The theor. results to date indicate that the A 2E'' state of NO3 undergoes a relatively strong JT distortion which may require inclusion of higher order vibronic couplings. The intensity of this transition may involve multiple intensity borrowing mechanisms via PJT coupling among the ∼X, ∼A, and ∼B states.
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36. 36
  Stanton, J. F. On the Vibronic Level Structure in the NO₃ Radical. I. The Ground Electronic State. J. Chem. Phys. 2007, 126, 134309, DOI: 10.1063/1.2715547
  36
  On the vibronic level structure in the NO3 radical. I. The ground electronic state
  Stanton, John F.
  Journal of Chemical Physics (2007), 126 (13), 134309/1-134309/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The model Hamiltonian approach of Koppel [etal] [Adv. Chem. Phys. 57, 59(1984)] was used to analyze the electronic spectroscopy of the nitrate radical (NO3). Simulations of neg. ion photodetachment of NO3-, the ~X2A'2←~B2E' dispersed fluorescence spectrum of NO3, and the ~B2E←~X2A'2 absorption spectrum are all in qual. agreement with expt., indicating that the model Hamiltonian contains most or all of the essential physics that govern the strongly coupled ~X2A'2 and ~B2E' electronic states of the radical. All 14 bands seen in the dispersed fluorescence spectrum <2600 cm-1 are assigned based on the simulations, filling in a few gaps left by previous work, and 7 addnl. bands <4000 cm-1 are tentatively assigned. The assignment is predicated on the assumption that the ν3 level of NO3 is ∼1000 cm-1 rather than 1492 cm-1 as is presently believed. Support for this reassignment (which assocs. the 1492 cm-1 band with the ν1+ν4 level) comes from both the model Hamiltonian spectrum and a FTIR feature at 2585 cm-1 that is consistent with the large and pos. cross anharmonicity between ν1 and ν4 needed for the alternative 1492 cm-1 assignment. An apparent systematic deficiency exists in the treatment of the model Hamiltonian for levels involving ν4. A discussion of the correlation between energy levels in the rigid D3h and C2v limits is illustrative, and provides insight into just how hard it is to treat the degenerate bending coordinate (q4) of NO3 accurately.
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37. 37
  Mukherjee, B.; Mukherjee, S.; Sardar, S.; Shamasundar, K. R.; Adhikari, S. An Ab Initio Investigation of Non-Adiabatic Couplings and Conical Intersections Among the Lowest Five Electronic States of the NO₃ Radical. Mol. Phys. 2017, 115, 2833– 2848, DOI: 10.1080/00268976.2017.1340680
  37
  An ab initio investigation of non-adiabatic couplings and conical intersections among the lowest five electronic states of the NO3 radical
  Mukherjee, Bijit; Mukherjee, Saikat; Sardar, Subhankar; Shamasundar, K. R.; Adhikari, Satrajit
  Molecular Physics (2017), 115 (21-22), 2833-2848CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
  In order to explore spectroscopic properties of a floppy mol., it is necessary to compute accurate diabatic potential energy surfaces (PESs) around the Franck-Condon region of nuclear configuration space. In such cases, it is desirable to include the non-adiabatic coupling terms (NACTs) through Jahn-Teller (JT) and pseudo Jahn-Teller (PJT) effects as well as other accidental conical intersections (CIs) prevailing among the electronic states of the mol. In this work, we investigate NACTs among the five lowest doublet electronic states of NO3 radical [ X2A'2(12B2), A2E''(12A2 and 12B1), B2E'(12A1 and 22B2)], in which all those effects (JT, PJT and CIs) have prominent contributions. The study is carried out in terms of normal modes (Qis) of NO3 and the CIs (JT and accidental) are located in a plane of nuclear configuration space of two such coordinates (Qi and Qj). We have calcd. ab initio adiabatic PESs and NACTs of those five states of NO3 at the MRCI level as functions of pairwise Qis. Further, to explore the nature of a given CI in terms of normal coordinates contributing to NACTs within a particular plane Qi-Qj, we have performed CP-MCSCF study computing analytic NACTs and characterised their contributions along various Qks.
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38. 38
  Mukherjee, B.; Mukherjee, S.; Sardar, S.; Shamasundar, K. R.; Adhikari, S. A Beyond Born-Oppenheimer Treatment of Five State Molecular System NO₃ and the Photodetachment Spectra of its Anion. Chem. Phys. 2018 (manuscript in press). DOI: 10.1016/j.chemphys.2018.09.017
  There is no corresponding record for this reference.
39. 39
  Köppel, H.; Cederbaum, L. S.; Domcke, W. Interplay of Jahn-Teller and pseudo-Jahn-Teller vibronic dynamics in the benzene cation. J. Chem. Phys. 1988, 89, 2023– 2040, DOI: 10.1063/1.455100
  39
  Interplay of Jahn-Teller and pseudo-Jahn-Teller vibronic dynamics in the benzene cation
  Koppel, H.; Cederbaum, L. S.; Domcke, W.
  Journal of Chemical Physics (1988), 89 (4), 2023-40CODEN: JCPSA6; ISSN:0021-9606.
  The static and dynamic aspects of the vibronic interaction of the ~B2E2g and ~C2A2u electronic states of PhH+ were analyzed. In the approxn. of linear vibrational and vibronic coupling, the model Hamiltonian for this system comprises 8 nonseparable vibrational modes, 6 of which are degenerate (2 of A1g symmetry, 4 of E2g symmetry, and 2 of E2u symmetry). The coupling consts. are estd. from existing ab initio SCF and semiempirical (CNDO/S) calcns. The topol. of the adiabatic potential-energy surfaces of this class of model Hamiltonians was investigated. The model exhibits a variety of conical intersections which dominate the vibronic dynamics. The dynamic problem was solved with simultaneous inclusion of 6 vibrational modes, 4 of which are degenerate (the Jahn-Teller coupling of 2 of the E2g modes is negligible). Hamiltonian matrixes with dimensions up to 6 × 106 are diagonalized using the Lanczos algorithm. After some adjustments of coupling consts., the calcn. reproduced the complex structure of the overlapping ~B2E2g-~C2A2u bands in the UPS of PhH. The vibronic structure of the lower-energy E2g band is dominated by a 2-mode Jahn-Teller effect in the ~B state. At higher energy, the marked diffuseness of the ~C2A2u band is a consequence of complete vibronic mixing with the lower-lying ~B state.
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40. 40
  Köppel, H. New Ultrafast Nonradiative Decay Mechanism in the Benzene Radical Cation. Chem. Phys. Lett. 1993, 205, 361– 370, DOI: 10.1016/0009-2614(93)87135-p
  There is no corresponding record for this reference.
41. 41
  Döscher, M.; Köppel, H. Multiple surface intersections and strong nonadiabatic coupling effects between the D̃²E_1u and Ẽ²B_2u states of C₆H₆⁺. Chem. Phys. 1997, 225, 93– 105, DOI: 10.1016/s0301-0104(97)00228-0
  There is no corresponding record for this reference.
42. 42
  Döscher, M.; Köppel, H.; Szalay, P. G. Multistate Vibronic Interactions in the Benzene Radical Cation. I. Electronic Structure Calculations. J. Chem. Phys. 2002, 117, 2645– 2656, DOI: 10.1063/1.1491397
  42
  Multistate vibronic interactions in the benzene radical cation. I. Electronic structure calculations
  Doscher, Martina; Koppel, Horst; Szalay, Peter G.
  Journal of Chemical Physics (2002), 117 (6), 2645-2656CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The multistate vibronic interactions in the benzene radical cation are investigated theor., employing the framework of a linear vibronic coupling scheme. The five lowest electronic states are included in the treatment; in view of the degeneracy of some states, this amts. to eight coupled potential energy surfaces. Different types of ab initio calcns. have been performed for the system parameters and been found to be in good mutual agreement, thus supporting each other. The calcns. reveal a whole sequence of low-energy conical intersections between the potential energy surfaces of different states. Their importance for the nuclear dynamics in this prototypical org. radical cation is pointed out. Wave-packet dynamical simulations for these coupled potential energy surfaces will be presented in the following paper (Paper II).
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43. 43
  Köppel, H.; Döscher, M.; Bâldea, I.; Meyer, H.-D.; Szalay, P. G. Multistate Vibronic Interactions in the Benzene Radical Cation. II. Quantum Dynamical Simulations. J. Chem. Phys. 2002, 117, 2657– 2671, DOI: 10.1063/1.1491398
  43
  Multistate vibronic interactions in the benzene radical cation. II. Quantum dynamical simulations
  Koppel, H.; Doscher, M.; Baldea, I.; Meyer, H.-D.; Szalay, P. G.
  Journal of Chemical Physics (2002), 117 (6), 2657-2671CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The multistate vibronic dynamics in the ~X 2E1g-~E 2B2u electronic states of the benzene radical cation was studied theor. by an ab initio quantum-dynamical approach. The vibronic coupling scheme and the ab initio values of the system parameters are adopted from the previous Paper I. Vibronic line spectra were obtained with the Lanczos procedure. Extensive calcns. on wave-packet propagation were performed with the aid of the multiconfiguration time-dependent Hartree method. Up to five coupled electronic potential energy surfaces and 13 vibrational degrees of freedom were included in these calcns. As a result, the impact of a 3rd electronic state (~X or ~B) on a strongly coupled manifold (~B-~C or D~-E~ states) is quant. assessed. It leads to a restructuring of the spectral envelope which is stronger for the ~B-D~-E~ than for the X~-B~-C~ system. The internal conversion dynamics was characterized by a stepwise transfer of electronic population to the lowest electronic state on a time scale of ∼100 fs, if the system was prepd. initially on the highest potential energy surface. Companion calcns. also were performed for the case when the system was prepd. in the intermediate state at t = 0; they show a branching of the electronic populations. These are all novel findings which are discussed in terms of conical intersections between the various potential energy surfaces. The importance of such multistate vibronic interactions for the photophysics and photochem. of medium-sized systems is pointed out.
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44. 44
  Köppel, H.; Bâldea, I.; Szalay, P. G. Combined Jahn-Teller and Pseudo-Jahn-Teller Effects in the Benzene Radical Cation. Advances in Quantum Chemistry; Academic Press, 2003; Vol. 44, pp 199– 217.
  There is no corresponding record for this reference.
45. 45
  Sardar, S.; Paul, A. K.; Sharma, R.; Adhikari, S. A “Classical Trajectory Driven Nuclear Dynamics by a Parallelized Quantum-Classical Approach to a Realistic Model Hamiltonian of Benzene Radical Cation. Int. J. Quantum Chem. 2009, 111, 2741– 2759, DOI: 10.1002/qua.22578
  There is no corresponding record for this reference.
46. 46
  Sardar, S.; Paul, A. K.; Sharma, R.; Adhikari, S. The Multistate Multimode Vibronic Dynamics of Benzene Radical Cation with a Realistic Model Hamiltonian Using a Parallelized Algorithm of the Quantumclassical Approach. J. Chem. Phys. 2009, 130, 144302, DOI: 10.1063/1.3108488
  46
  The multistate multimode vibronic dynamics of benzene radical cation with a realistic model Hamiltonian using a parallelized algorithm of the quantumclassical approach
  Sardar, Subhankar; Paul, Amit Kumar; Sharma, Rahul; Adhikari, Satrajit
  Journal of Chemical Physics (2009), 130 (14), 144302/1-144302/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We demonstrate the workability of a parallelized algorithm of the time-dependent discrete variable representation (TDDVR) method to explore the detailed dynamical aspects of vibronic interaction in two three-state model Hamiltonians (X 2E1g, B 2E2g, C 2A2u and B 2E2g, D 2E1u, E 2B2u) of benzene radical cation along with a preliminary investigation on its five electronic states (X 2E1g, B 2E2g, C 2A2u, D 2E1u, and E2B2u). Since those electronic states are interconnected through a series of conical intersections, we have used six and nine vibronically important modes for the three- and five-state Hamiltonians, resp., in order to perform the quantum dynamics on such system. The population profiles calcd. by using our TDDVR approach show reasonably good agreement with the results obtained by exact quantum mech. (multiconfiguration time-dependent Hartree) method, whereas the corresponding (calcd.) photoabsorption spectra originating from various electronic states agree well with the exptl. ones. It is important to note that the parallelized algorithm of our TDDVR approach reduces the computation cost by more than an order of magnitude compared to its serial analog. The TDDVR approach appears to be a good compromise between accuracy and speed for such large mol. system, where quantum mech. description is needed in a restricted region. (c) 2009 American Institute of Physics.
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47. 47
  Sardar, S.; Adhikari, S. A quantum-classical simulation of a multi-surface multi-mode nuclear dynamics on C₆H₆⁺ incorporating degeneracy among electronic states. J. Chem. Sci. 2012, 124, 51– 58, DOI: 10.1007/s12039-011-0195-z
  47
  A quantum-classical simulation of a multi-surface multi-mode nuclear dynamics on incorporating degeneracy among electronic states
  Sardar, Subhankar; Adhikari, Satrajit
  Journal of Chemical Sciences (Bangalore, India) (2012), 124 (1), 51-58CODEN: JCSBB5; ISSN:0974-3626. (Indian Academy of Sciences)
  We have performed a nuclear dynamics simulation to calc. photoelectron spectra and population profiles of benzene radical cation (C6H) employing the parallelized Time Dependent Discrete Variable Representation (TDDVR) approach. For this purpose, we have considered two multi-state multi-mode model Hamiltonians of C6H with degeneracy among the eectronic states: (a) One consists of three states and eight modes, which in turn leads to a five state thirteen mode Hamiltonian (X 2E1g -B 2E2g -C 2A2u ) due to the degeneracy; and (b) The other is constituted of three states and thirteen modes which is basically a five state twenty mode Hamiltonian (B 2E2g -D 2E1u -E 2B2u ) for the same reason. Since these electronic states are interconnected by several conical intersections in the vicinity of the Franck Condon region, it will be challenging to pursue such large dynamical calcn. in the presence of nonadiabaticity among the electronic states. The spectral as well as population profiles calcd. with the advent of TDDVR approach show reasonably good agreement with the results obtained by the Multi Configuration Time Dependent Hartree (MCTDH) methodol.
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48. 48
  Goode, J. G.; Hofstein, J. D.; Johnson, P. M. The observation of strong pseudo-Jahn-Teller activity in the benzene cation B̃²E_2g state. J. Chem. Phys. 1997, 107, 1703– 1716, DOI: 10.1063/1.474526
  There is no corresponding record for this reference.
49. 49
  Johnson, P. M. The Jahn-Teller effect in the lower electronic states of benzene cation. II. Vibrational analysis and coupling constants of the B̃²E_2g state. J. Chem. Phys. 2002, 117, 10001– 10007, DOI: 10.1063/1.1519007
  There is no corresponding record for this reference.
50. 50
  Johnson, P. M. The Jahn-Teller Effect in the Lower Electronic States of Benzene Cation. I. Calculation of Linear Parameters for the e_2g Modes. J. Chem. Phys. 2002, 117, 9991– 10000, DOI: 10.1063/1.1519006
  50
  The Jahn-Teller effect in the lower electronic states of benzene cation. I. Calculation of linear parameters for the e2g modes
  Johnson, Philip M.
  Journal of Chemical Physics (2002), 117 (22), 9991-10000CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Jahn-Teller (JT) coupling parameters can be simply derived from features of the potential energy surfaces of the JT active vibrational modes of a mol. subject to that effect. Potential energy curves representing cuts of D2h symmetry have been calcd. using d. functional theory for each of the four active e2g modes of benzene cation in each of the lowest three degenerate electronic states. From these curves and the vibrational frequencies, the JT coupling parameters can be found (vibrational numbering follows Wilson's notation, coupling parameters are in Moffit's notation). For the C6H+6X 2E1g state the largest JT coupling parameter is provided by mode 6 (D=0.48), followed closely by mode 8 (0.30) and then mode 9 (0.10). Mode 7 provides almost no stabilization. For the B 2E2g state, calcd. coupling parameters are very large, particularly for modes 8 (1.36) and 6 (0.93). Modes 7 (0.10) and 9 (0.07) are smaller but finite. For the D 2E1u state, mode 6 has an impressive linear coupling parameter of 4.12, able to support several vibrations below the JT cusp. Indications of a substantial quadratic stabilization for that mode result in the possibility of a completely static distortion in that state. Similar, but not identical, results are obtained for C6D6+.
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51. 51
  Ghosh, S.; Mukherjee, S.; Mukherjee, B.; Mandal, S.; Sharma, R.; Chaudhury, P.; Adhikari, S. Beyond Born-Oppenheimer theory for ab initio constructed diabatic potential energy surfaces of singlet H₃⁺ to study reaction dynamics using coupled 3D time-dependent wave-packet approach. J. Chem. Phys. 2017, 147, 074105, DOI: 10.1063/1.4998406
  51
  Beyond Born-Oppenheimer theory for ab initio constructed diabatic potential energy surfaces of singlet H3+ to study reaction dynamics using coupled 3D time-dependent wave-packet approach
  Ghosh, Sandip; Mukherjee, Saikat; Mukherjee, Bijit; Mandal, Souvik; Sharma, Rahul; Chaudhury, Pinaki; Adhikari, Satrajit
  Journal of Chemical Physics (2017), 147 (7), 074105/1-074105/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The workability of beyond Born-Oppenheimer theory to construct diabatic potential energy surfaces (PESs) of a charge transfer atom-diatom collision process has been explored by performing scattering calcns. to ext. accurate integral cross sections (ICSs) and rate consts. for comparison with most recent exptl. quantities. We calc. non-adiabatic coupling terms among the lowest three singlet states of H3 + system (11A', 21A', and 31A') using MRCI level of calcn. and solve the adiabatic-diabatic transformation equation to formulate the diabatic Hamiltonian matrix of the same process [S. Mukherjee et al., J. Chem. Phys. 141, 204306 (2014)] for the entire region of nuclear configuration space. The nonadiabatic effects in the D+ + H2 reaction has been studied by implementing the coupled 3D time-dependent wave packet formalism in hyperspherical coordinates [S. Adhikari and A. J. C. Varandas, Comput. Phys. Commun. 184, 270 (2013)] with zero and non-zero total angular momentum (J) on such newly constructed accurate (ab initio) diabatic PESs of H3+. We have depicted the convergence profiles of reaction probabilities for the reactive non-charge transfer, non-reactive charge transfer, and reactive charge transfer processes for different collisional energies with respect to the helicity (K) and total angular momentum (J) quantum nos. Finally, total and state-to-state ICSs are calcd. as a function of collision energy for the initial rovibrational state (v = 0, j = 0) of the H2 mol., and consequently, those quantities are compared with previous theor. and exptl. results. (c) 2017 American Institute of Physics.
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52. 52
  Werner, H.-J.; MOLPRO , version 2010.1, a Package of Ab initio Programs, 2010; see http://www.molpro.net.
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Topological Effects in Vibronically Coupled Degenerate Electronic States: A Case Study on Nitrate and Benzene Radical CationClick to copy article linkArticle link copied!

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Abstract

1. Introduction

2. BBO Theory and Diabatic Hamiltonian

3. Dressed Adiabatic, Dressed Diabatic, and Adiabatic-via-Dressed Diabatic PECs: Topological Effects

4. Results and Discussion

4.1. NO3 Radical

Figure 1

Figure 2

Figure 3

Figure 4

4.2. Bz+ Radical Cation

Figure 5

Figure 6

Figure 7

Figure 8

5. Conclusions

Author Information

Acknowledgments

References

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Abstract

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

References

Topological Effects in Vibronically Coupled Degenerate Electronic States: A Case Study on Nitrate and Benzene Radical Cation
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4.1. NO₃ Radical

4.2. Bz⁺ Radical Cation