An Exhaustive Quantum-Classical Study of C6F6+ Using the Newly Formulated Parallel TDDVR Method.

We recently implemented our parallelized quantum-classical dynamical approach, known as the Time-Dependent Discrete Variable Representation (TDDVR) method, which is applied to the spectroscopically important hexafluorobenzene (HFBz) radical cation, where several conical intersections exist in their seven lowest excited electronic states (S11B2u, S21E1g, S31B1u, S41E1u, and S51A2u) considering degeneracy among potential energy surfaces (PESs), to demonstrate their various dynamical aspects. This new parallel version shows almost linear scalability with increasing number of computing processors. To get photoelectron (PE) spectra, Mass-Analyzed Threshold Ionization (MATI) spectra, population dynamics, and many other dynamical observables, the first-principles dynamics is applied at the state-of-the-art level to the corresponding Hamiltonian, where the Jahn-Teller (JT) and pseudo-Jahn-Teller (PJT) type interactions are involved in those coupled seven electronic states. The quantum-classical method is used to thoroughly analyze the effects of these couplings on the nuclear dynamics of the involved electronic states, and the findings are compared with those observables obtained from experiments. Intrinsic dynamical properties are explained using the reduced densities of the wave packet (WP) in a coupled electronic manifold. The PE and MATI spectra of HFBz computed using TDDVR are found to be in good agreement with earlier experimental data and other theoretically simulated spectra.

Construction of Beyond Born-Oppenheimer Based Diabatic Surfaces and Generation of Photoabsorption Spectra: The Touchstone Pyrazine (C4 N2 H4 ).

We construct theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) based diabatic potential energy surfaces (PESs) of pyrazine (C4 N2 H4 ) molecule involving lowest four excited adiabatic PESs (S1 to S4 ) and nonadiabatic coupling terms (NACTs) among those surfaces as functions of nonadiabatically active normal modes (Q1 , Q6a , Q9a and Q10a ) to compute its photoabsorption (PA) spectra. Those adiabatic PESs are calculated using CASSCF as well as MRCI based methodologies, where NACTs are obtained from CP-MCSCF approach. Employing ab initio quantities (adiabatic PESs and NACTs), it is possible to depict the conical intersections (CIs) and develop matrices of diabatic PESs over six normal mode planes. Once single-valued, smooth, symmetric and continuous 2×2 and 4×4 diabatic surface matrices are in hand for the first time, such matrices are used to perform multi-state multi-mode nuclear dynamics with the aid of Time-Dependent Discrete Variable Representation (TDDVR) methodology initializing the product type wavefunction on 1 1 B 1 u ${{1}^{1}{B}_{1u}}$ (S1 ) and 1 1 B 2 u ${{1}^{1}{B}_{2u}}$ (S2 ) states to obtain the corresponding PA spectra. TDDVR calculated spectra for those states (S1 and S2 ) obtained from BBO based 2×2 and 4×4 diabatic surface matrices show good and better agreement with the experimental results, respectively. Both of these calculated results depict better peak progression over the existing profiles of Multi-Configuration Time-Dependent Hartree (MCTDH) dynamics over 2×2 Vibronic Coupling Model (VCM) Hamiltonian.

Beyond Born-Oppenheimer based diabatic surfaces of 1,3,5-C6H3F3+ to generate the photoelectron spectra using time-dependent discrete variable representation approach.

In this article, Beyond Born-Oppenheimer (BBO) treatment is implemented to construct diabatic potential energy surfaces (PESs) of 1,3,5-C6H3F3+ over a series [eighteen (18)] of two-dimensional (2D) nuclear planes constituted with eleven normal modes (Q2, Q9x, Q9y, Q13x, Q13y, Q18x, Q18y, Q10x, Q10y, Q12x and Q12y) to include all possible nonadiabatic interactions among six coupled electronic states (X̃2E'', , B̃2E' and ). We had formulated explicit expressions of adiabatic to diabatic transformation (ADT) equations [S. Mukherjee, J. Dutta, B. Mukherjee, S. Sardar and S. Adhikari, J. Chem. Phys., 2019, 150, 064308] for the same system forming six state sub-Hilbert space and at present, these ADT equations are solved by incorporating MRCI level ab initio adiabatic PESs and CP-MCSCF calculated nonadiabatic coupling terms (NACTs) to derive diabatic PESs and couplings. Such single-valued, smooth, symmetric and continuous diabatic surface matrices are utilized to carry out multi-state multi-mode nuclear dynamics with the help of time-dependent discrete variable representation (TDDVR) methodology to compute the photoelectron (PE) spectra of 1,3,5-C6H3F3. Our theoretically calculated spectra for X̃2E'', and states using BBO treatment and TDDVR dynamics show peak by peak correspondence with the experimental results as well as better than the findings of the multi-configuration time-dependent Hartree (MCTDH) method.

A beyond Born-Oppenheimer treatment of C6H6 + radical cation for diabatic surfaces: Photoelectron spectra of its neutral analog using time-dependent discrete variable representation.

We employ theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation (C6H6 +) for the first time and explore the workability of the time-dependent discrete variable representation (TDDVR) method for carrying out dynamical calculations to evaluate the photoelectron (PE) spectra of its neutral analog. Ab initio adiabatic PESs and nonadiabatic coupling terms are computed over a series of pairwise normal modes, which exhibit rich nonadiabatic interactions starting from Jahn-Teller interactions and accidental conical intersections/seams to pseudo Jahn-Teller couplings. Once the electronic structure calculation is completed on the low-lying five doublet electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of the cationic species, diabatization is carried out employing the adiabatic-to-diabatic transformation (ADT) equations for the five-state sub-Hilbert space to compute highly accurate ADT angles, and thereby, single-valued, smooth, symmetric, and continuous diabatic PESs and couplings are constructed. Subsequently, such surface matrices are used to perform multi-state multi-mode nuclear dynamics for simulating PE spectra of benzene. Our theoretical findings clearly depict that the spectra for X̃2E1g and B̃2E2g-C̃2A2u states obtained from BBO treatment and TDDVR dynamics exhibit reasonably good agreement with the experimental results as well as with the findings of other theoretical approaches.

Effects of site-specific substitution to hexacene and its effect towards singlet fission.

The singlet fission (SF) process, which is involved in organic solar cell, is a spin allowed and extremely fast internal conversion process by which a photo-excited singlet exciton produces two triplets. To accelerate this fission and to increase the efficiency of solar cell, designing of new materials/molecules is an interesting area of research and our current interest. Several alternate hydrocarbons of the acene series and their derivatives with structural variety are desirable for this purpose. Therefore, we have theoretically modeled and investigated different substituted hexacenes in detail. Different electron donating groups (EDGs) as well as electron withdrawing groups (EWGs) and few groups of other varieties are chosen for site-specific monosubstitution to the hexacene ring for manipulation of their excited singlet-triplet energy levels and thereby to match the SF driving force (Δ) and triplet-triplet annihilation (TTA) deactivation force (Ω). The geometries, electronic structures, frontier molecular orbital (FMO) energies, optimization of excited state and calculation of Δ and Ω of the substituted hexacenes are investigated with Time Dependent Density functional theory (TDDFT) method using B3LYP/6-31G∗ basis set. After a detail theoretical analysis on sixty five (65) hexacene derivatives, the δ-triisopropylsilylethynyl (-TIPS) hexacene and ß-NO2 hexacene are predicted to exhibit good SF characteristics.

ADT: A Generalized Algorithm and Program for Beyond Born-Oppenheimer Equations of "N" Dimensional Sub-Hilbert Space.

The major bottleneck of first principle based beyond Born-Oppenheimer (BBO) treatment originates from large number and complicated expressions of adiabatic to diabatic transformation (ADT) equations for higher dimensional sub-Hilbert spaces. In order to overcome such shortcoming, we develop a generalized algorithm, "ADT" to generate the nonadiabatic equations through symbolic manipulation and to construct highly accurate diabatic surfaces for molecular processes involving excited electronic states. It is noteworthy to mention that the nonadiabatic coupling terms (NACTs) often become singular (removable) at degenerate point(s) or along a seam in the nuclear configuration space (CS) and thereby, a unitary transformation is required to convert the kinetically coupled (adiabatic) Hamiltonian to a potentially (diabatic) one to avoid such singularity(ies). The "ADT" program can be efficiently used to (a) formulate analytic functional forms of differential equations for ADT angles and diabatic potential energy matrix and (b) solve the set of coupled differential equations numerically to evaluate ADT angles, residue due to singularity(ies), ADT matrices, and finally, diabatic potential energy surfaces (PESs). For the numerical case, user can directly provide ab initio data (adiabatic PESs and NACTs) as input files to this software or can generate those input files through in-built python codes interfacing MOLPRO followed by ADT calculation. In order to establish the workability of our program package, we selectively choose six realistic molecular species, namely, NO2 radical, H3+, F + H2, NO3 radical, C6H6+ radical cation, and 1,3,5-C6H3F3+ radical cation, where two, three, five and six electronic states exhibit profound nonadiabatic interactions and are employed to compute diabatic PESs by using ab initio calculated adiabatic PESs and NACTs. The "ADT" package released under the GNU General Public License v3.0 (GPLv3) is available at https://github.com/AdhikariLAB/ADT-Program and also as the Supporting Information of this article.

Conical intersections and nonadiabatic coupling terms in 1,3,5-C6H3F3 +: A six state beyond Born-Oppenheimer treatment.

In order to circumvent numerical inaccuracy originating from the singularity of nonadiabatic coupling terms (NACTs), we need to perform kinetically coupled adiabatic to potentially coupled diabatic transformation of the nuclear Schrödinger Equation. Such a transformation is difficult to achieve for higher dimensional sub-Hilbert spaces due to inherent complicacy of adiabatic to diabatic transformation (ADT) equations. Nevertheless, detailed expressions of ADT equations are formulated for six coupled electronic states for the first time and their validity is extensively examined for a well-known radical cation, namely, 1,3,5-C6H3F3 + (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states (X̃2E'', Ã2A2 '', B̃2E', and C̃2A2 '), where several types of nonadiabatic interactions, like Jahn-Teller conical intersections (CI), accidental CIs, accidental seams (series of degenerate points), and pseudo Jahn-Teller interactions can be observed over the Franck-Condon region of nuclear configuration space. Those interactions are depicted by exploring degenerate components of C-C asymmetric stretching, C-C symmetric stretching, and C-C-C scissoring motion (Q9x, Q9y, Q10x, Q10y, Q12x, and Q12y) to compute complete active space self-consistent field level adiabatic PESs and NACTs as implemented in the MOLPRO quantum chemistry package. Subsequently, we perform the ADT using our newly devised fifteen (15) ADT equations to locate the position of CIs, verify the quantization of NACTs, and to construct highly accurate diabatic PESs.

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

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. A 2017, 121, 6314-6326]. In the case of NO3 radical, the nonadiabatic coupling terms (NACTs) among the low-lying five electronic states, namely, X̃ 2A2 ' (12B2), A~ 2E″ (12A2 and 12B1), and B~ 2E' (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.

Monitoring of the energy levels by heteroatom substitution to hexacene and controlling over singlet fission and photo-oxidative resistance.

The singlet fission is a spin allowed and extremely fast internal conversion process involved in solar cell by which a photo-excited singlet exciton is splitted into two triplet ones. For effective singlet fission and to increase the efficiency of solar cell, designing of new molecules is an interesting area of research and our current interest. The silicon substituted oligocenes, commonly known as silaoligocenes, are found to be the efficient singlet fission material due to their special characteristics. We have shown the SF energy criteria satisfied by the singlet and triplet states of various silahexacene derivatives, and theoretically predicted whether such molecules exhibit fission properties or not. The fluorine atoms have been substituted to various positions of different silahexacenes to manipulate their singlet and triplet energy levels. As fluorine being the most electro-negative substituent, it is capable of lowering frontier molecular orbital energies effectively. Thus, the material can easily match SF energy criteria to compute the SF driving force or triplet-triplet annihilation possibility. The geometries, electronic structures, frontier molecular orbital energies, optimization of excited state and calculation of energies associated with fission process of the substituted hexacene are investigated with well known quantum mechanical methods.

Ab initio constructed diabatic surfaces of NO2 and the photodetachment spectra of its anion.

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

Multisurface multimode molecular dynamical simulation of naphthalene and anthracene radical cations by using nearly linear scalable time-dependent discrete variable representation method.

The major portion of the algorithm of the time-dependent discrete variable representation (TDDVR) method is recently parallelized using the shared-memory parallelization scheme with the aim of performing dynamics on relatively large molecular systems. Because of the astronomical importance of naphthalene and anthracene, we have investigated their radical cations as models for theoretical simulation of complex photoelectron spectra and nonradiative decay process using the newly implemented parallel TDDVR code. The strong vibronic coupling among the six lowest doublet electronic states makes these polynuclear hydrocarbons dynamically important. The aim of the present investigation is to show the efficiency of our current TDDVR algorithm to perform dynamics on large dimensional quantum systems in vibronically coupled electronic manifold. Both the sequential and the parallelized TDDVR algorithms are almost linear scalable for an increase in number of processors. Because a significant speed-up is achieved by cycling in the correct way over arrays, all of the simulations are performed within a reasonable wall clock time. Our theoretical spectra well reproduce the features of the corresponding experimental analog. The dynamical outcomes, for example, population, photoelectron spectra, and diffused interstellar bands, etc., of our quantum-classical approach show good agreement with the findings of the well-established quantum dynamical method, that is, multi configuration time-dependent Hartree (MCTDH) approach.

Multi-state multi-mode nuclear dynamics on three isomers of C6H4F2+ using parallelized TDDVR approach.

We have performed molecular dynamics on the three isomers of the difluorobenzene radical cation (C(6)H(4)F(2)(+)) after excitation from the ground state to a specific higher electronically excited state by using our recently implemented parallelized Time-Dependent Discrete Variable Representation (TDDVR) methodology. A five-state eleven-mode realistic model Hamiltonian for o-C(6)H(4)F(2)(+) and two separate five-state ten-mode Hamiltonians for m- and p-isomer of the same radical cation are considered, where those five electronic states are interconnected through several conical intersections in the vicinity of the Franck-Condon (FC) region and thus the dynamics for each case become complex. The photoelectron, mass analyzed threshold ionization spectra and population profiles obtained by using our TDDVR approach show reasonably good agreement with the results obtained by multiconfiguration time dependent Hartree (MCTDH) method. It is worthwhile to mention that the parallelized TDDVR algorithm reduces the computation time by more than an order of magnitude compared to its serial analog and, therefore, such approach appears to be a good compromise between accuracy and speed for a large molecular system.

Effect of surface modes on the six-dimensional molecule-surface scattering dynamics of H2-Cu(100) and D2-Cu(111) systems.

We include the phonon modes originating from the three layers of Cu(100)/Cu(111) surface atoms on the dynamics of molecular [H(2)(v,j)/D(2)(v,j)] degrees of freedom (DOFs) through a mean field approach, where the surface temperature is incorporated into the effective Hamiltonian (potential) either by considering Boltzmann probability (BP) or by including the Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes. The formulation of effective potential has been carried out by invoking the expression of transition probabilities for phonon modes known from the "stochastic" treatment of linearly forced harmonic oscillator (LFHO). We perform four-dimensional (4D⊗2D) as well as six-dimensional (6D) quantum dynamics on a parametrically time and temperature-dependent effective Hamiltonian to calculate elastic/inelastic scattering cross-section of the scattered molecule for the H(2)(v,j)-Cu(100) system, and dissociative chemisorption-physisorption for both H(2)(v,j)-Cu(100) and D(2)(v,j)-Cu(111) systems. Calculated sticking probabilities by either 4D⊗2D or 6D quantum dynamics on an effective potential constructed by using BP factor for the initial state distribution of the phonon modes could not show any surface temperature dependence. In the BEP case, (a) both 4D⊗2D and 6D quantum dynamics demonstrate that the phonon modes of the Cu(100) surface affect the state-to-state transition probabilities of the scattered H(2) molecule substantially, and (b) the sticking probabilities due to the collision of H(2) on Cu(100) and D(2) on Cu(111) surfaces show noticeable and substantial change, respectively, as function of surface temperature only when the quantum dynamics of all six molecular DOFs are treated in a fully correlated manner (6D).

The effect of phonon modes on the H2(v, j)/D2(v, j)-Cu(1nn) scattering processes.

We include the effect of the phonon modes originating from the three layers of Cu(1nn) surface atoms on the dynamics of incoming molecular [H(2)(v, j)/D(2)(v, j)] degrees of freedom (DOFs) through a mean-field approach, where the surface temperature is incorporated into the effective potential by considering Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes calculated within harmonic approximation. Such time and temperature dependent effective Hamiltonian is further subdivided assuming a weak coupling between the two sets of molecular DOFs, namely, (x, y, z, Z) and (X, Y), respectively, in particular, to reduce the computational cost and the corresponding coupled quantum dynamical equations of motion have been formulated in terms of Time Dependent Discrete Variable Representation (TDDVR) approach. We demonstrate the workability of TDDVR method to investigate the scattering of H(2)(v, j) on Cu(1nn) surface by calculating the reaction probabilities and scattering cross-sections. Calculated results show that the phonon modes affect (a) the state-to-state transition probabilities of the scattered H(2) molecule substantially but chemisorption and physisorption processes negligibly and (b) the reaction probability of the incoming D(2) molecule noticeably.

Single surface beyond Born-Oppenheimer equation for a three-state model Hamiltonian of Na3 cluster.

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

The multistate multimode vibronic dynamics of benzene radical cation with a realistic model Hamiltonian using a parallelized algorithm of the quantumclassical approach.

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 (2)E(1g), B (2)E(2g), C (2)A(2u) and B (2)E(2g), D (2)E(1u), E (2)B(2u)) of benzene radical cation along with a preliminary investigation on its five electronic states (X (2)E(1g), B (2)E(2g), C (2)A(2u), D (2)E(1u), and E(2)B(2u)). Since those electronic states are interconnected through a series of conical intersections, we have used six and nine vibronically important modes for the three- and five-state Hamiltonians, respectively, in order to perform the quantum dynamics on such system. The population profiles calculated by using our TDDVR approach show reasonably good agreement with the results obtained by exact quantum mechanical (multiconfiguration time-dependent Hartree) method, whereas the corresponding (calculated) photoabsorption spectra originating from various electronic states agree well with the experimental ones. It is important to note that the parallelized algorithm of our TDDVR approach reduces the computation cost by more than an order of magnitude compared to its serial analog. The TDDVR approach appears to be a good compromise between accuracy and speed for such large molecular system, where quantum mechanical description is needed in a restricted region.

A quantum-classical approach to the molecular dynamics of butatriene cation with a realistic model Hamiltonian.

We are investigating the molecular dynamics of the butatriene cation after excitation from the ground state (X(2)B(2g)) to the first excited electronic state (A(2)B(2u)) by using the time-dependent discrete variable representation (TDDVR) method. The investigation is being carried out with a realistic 18-mode model Hamiltonian consisting of all the vibrational degrees of freedom of the butatriene molecule. First, we perform the simulation on a basic five mode model, and then by including additional thirteen modes as bath on the basic model. This sequential inclusion of bath modes demonstrates the effect of so called weak modes on the subsystem, where the calculations of energy and population transfer from the basic model to the bath quantify the same effect. The spectral profile obtained by using the TDDVR approach shows reasonably good agreement with the results calculated by the quantum mechanical approach/experimental measurement. It appears that the TDDVR approach for those large systems where quantum mechanical description is needed in a restricted region, is a good compromise between accuracy and speed.