Density dynamics in some quantum systems

Authors

  • Munmun Khatua,

    1. Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
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  • Debdutta Chakraborty,

    1. Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
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  • Pratim Kumar Chattaraj

    Corresponding author
    1. Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
    • Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
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Abstract

The quantum domain behavior of classical nonintegrable systems is well-understood by the implementation of quantum fluid dynamics and quantum theory of motion. These approaches properly explain the quantum analogs of the classical Kolmogorov–Arnold–Moser type transitions from regular to chaotic domain in different anharmonic oscillators. Field-induced tunneling and chaotic ionization in Rydberg atoms are also analyzed with the help of these theories. Quantum fluid density functional theory may be used to understand different time-dependent processes like ion-atom/molecule collisions, atom-field interactions, and so forth. Regioselectivity as well as confined atomic/molecular systems and their reactivity dynamics have also been explained. © 2013 Wiley Periodicals, Inc.

Introduction

Ever since Dirac announced in 1929 that “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble,” scientists have been trying to develop efficient numerical strategies to solve the Schrödinger equation for atoms and molecules which allowed quantum chemistry to become an individual scientific discipline. To have a classical interpretation of quantum mechanics, several researchers have been trying to propose different versions of quantum mechanics in three-dimensional (3D) space. Unlike the wave function which is a 3N dimensional function for an N-electron system, the single-particle density is a 3D function and any quantum theory based on the latter would help developing models, making direct connection with experiments, for example, using X-ray, apart from providing a considerable amount of reduction in computational labor. Modern density functional theory (DFT) and its different variants to tackle excited states and time-dependent (TD) situations have provided proper theoretical justification for all such density-based theories heuristically proposed earlier.

Just like the density functional scheme developed by Hohenberg, Kohn, and Sham was adequate for the ground state, a theory was needed to describe the excited states which are difficult to handle in DFT. The generalization of the Hohenberg-Kohn theorem to the excited states was given by Gunnarsson and Lundqvist1 through the thermodynamic version of spin density formalism. von Barth2 pointed out that the exchange correlation energy in their theory should be symmetry dependent; however, in actual calculation exchange correlational energy functional is approximated by the local density approximation (LDA) with no dependence in symmetry. This method hence failed as it does not reduce to a lone determinant.2 To solve this problem, Xα method was incorporated expressing the multiplet energy as sum of one-determinant energies. But all the above methods could not be extended past the lowest state with a given space and spin symmetry. A DFT method which would account for the general excited state was needed and hence Slater's transition state theory3 was a step forward in this direction.

The success of DFT for stationary systems has inspired its applications toward the TD situations. Bloch4 laid the foundations to a TDDFT in the form of TD version of the Thomas–Fermi (TF) theory. Runge and Gross5 showed that the mapping between the TD density and the TD potential is unique in general conditions. However, Xu and Rajagopal6 criticized this fact and indicated that not the TD potential to TD density but the TD potential to the TD current density mapping is uniquely invertible which was later proved wrong by Dhara and Ghosh.7 Another TDDFT for many electron systems is the quantum fluid DFT (QFDFT) which is an amalgamation of TDDFT (TDDFT) and quantum fluid dynamics (QFD). QFDFT has been successfully applied in studies of ion-atom/molecule collisions,8–10 atom–field interactions,11, 12 and dynamics of chemical reactivity parameters13–16 such as electronegativity, chemical hardness, entropy, and polarizability in chemical reactions.

This review essentially focuses on some representative examples through the study of chaos and nonlinear dynamical systems carried out by our group using the QFD and the QFDFT and also the study of the reactivity dynamics of various systems in free as well as confined environment.

Density Functional Theory

The substitution of the more complicated N-electron wave function Ψ(x1,x2,… … … xN) and the associated Schrödinger equation by the much simpler one electron density ρ(r) makes the DFT a remarkable one where,

equation image(1)

From the atomic model of Thomas and Fermi,17, 18 it is clear that the energy of an electronic system can be articulated in terms of the electron density ρ. Although being an approximate effort, it was widely applicable19 in atomic physics. DFT20–22 uses ρ(r) as a fundamental variable of an electronic structure theory for many electron systems namely, atoms, molecules, and solids where the single-particle density provides all the information regarding the system. This has been proved true by Hohenberg and Kohn,22 for the ground state or lowest symmetry state, in terms of their famous two theorems. The first theorem states that within a trivial additive constant, the one electron density ρ(r) determines the external potential v(r) for the nondegenerate ground state of an N-electron system. The inverse mapping between the two also exists. Electron density ρ determines the number of electrons of the system, the ground state wave function Ψ, and all other electronic properties of the system.

equation image(2)

The second theorem deals with the energy variational principle. A trial density ρ(r), which is not the exact ρ(r), will give an energy greater than the exact energy, E0.

The Hohenberg–Kohn variational principle is

equation image(3)

where μ is the Lagrange multiplier. Equation (3) can be simplified into

equation image(4)

The functional derivative of energy E with respect to ρ is represented by equation image.

The functional form of E[ρ] can be expressed as

equation image(5)

v(r) being the external potential of the system and F[ρ] is a universal functional comprising the kinetic energy T and the electron–electron interaction energy Eee. Therefore,

equation image(6)

The RHS of Eq. (6) comprises the kinetic energy functional T[ρ], the classical coulomb part of the electron–electron interaction energy Eee, and the nonclassical part of Eee, the exchange-correlation energy of the electrons, represented by Exc. Therefore, evaluation of T[ρ] and Exc[ρ] would give us the total energy functional E[ρ]. Considering homogeneity of the system, the kinetic energy functional T[ρ] can be obtained from the TF energy term as follows,

equation image(7)

where equation image. Considering inhomogeneity of the system, the Weizsäcker correction23 to the above term is as follows,

equation image(8)

The kinetic energy functional T[ρ] has the following form,

equation image(9)

where ∧ is shown to have a value of 1/9.

From the viewpoint of Ghosh and Deb,24 a local kinetic energy functional can be expressed as

equation image(10)

where equation image. For proper global and local behavior as well as a correct functional derivative, Chattaraj and Sengupta16 introduced another atomic kinetic energy functional, namely,

equation image(11)

where equation image and the constant a(N) depends on the number of electrons N. In dealing with diatomic molecules, the molecular kinetic energy functional may be written as16

equation image(12a)

where

equation image(12b)

and R is the internuclear distance.

The exchange-correlation potential Vxc, is the functional derivative of Exc with respect to ρ following the LDA,

equation image(13)

and

equation image(14)

the exchange part is given by the Dirac25 exchange energy functional, which shows a ρ4/3-dependence of the exchange energy. The correlation part is a logarithmic function of density.

Various approaches have been developed to solve for the density and the energy from this density equation, the most well-known being that of Kohn and Sham.26 Kohn and Sham formulation of DFT introduces a corresponding noninteracting reference system of N electrons in which each electron moves under the influence of an effective potential Veff, taking into account both the exchange and the correlation effects,

equation image(15)

or

equation image(16)

Hence, the Euler equation could be written as

equation image(17)

Using the condition,

equation image(18)

we get

equation image(19)

For a given Veff(r), the density ρ(r) can be obtained which satisfies eqs. (18) and (19) by solving the one-electron equation

equation image(20)

and setting

equation image(21)

The density ρ(r), the ground state energy E, the kinetic energy T, and other properties can now be obtained from the solution of the Kohn–Sham equations. The following equations,

equation image(22)

and

equation image(23)

are the expressions for E and T.

In conclusion, we can say that DFT plays an effective role in the study of atoms, molecules, and solids. Its applications in the nuclei problem as well as the liquid state have been quite successful. Some researchers27–29 have used DFT to study light as well as heavy atoms. Local spin DFT allows the study of open shell atoms. DFT has been used to study the physical properties of crystalline solids. Very large molecules like proteins30, 31 have also been studied by using DFT.

TDDFT of Many-Electron Systems

The study of the interaction of many-electron systems such as atoms, molecules, and solids with the TD external fields has been an important area of research over many years. In linear response regime, it is possible to obtain various physical quantities if we consider a small perturbation to the initial ground state of the system caused by the external field. But to study the interaction of many-electron systems with strong laser fields, we have to go beyond linear response to investigate the properties. An exact solution of TD Schrödinger equation is not within the reach of computational methods, therefore, to efficiently handle the TD many-electron problems, new approaches are to be explored, and one such efficient approach is the TDDFT.

The well-known TF theory17, 18 led to the DFT and a corresponding TD version of the TF theory was soon introduced by Bloch.4 The TD theory of Bloch is an approximate version of the more general TDDFT. Soon after the Schrödinger equation was proposed, the desire to have a classical interpretation of the quantum mechanics was so strong that its hydrodynamic analog was derived by Madelung.32 In later years, QFD33, 34 played a major role in the development of TDDFT. An important aspect in which the TDDFT differs from DFT is the inclusion of another density variable, the current density, along with the electron density as well.

The Hohenberg–Kohn-Sham version of DFT allows us to express the energy of an N-electron system under the influence of an external potential vN (r) due to the nuclei, as a functional of the single-particle electron density ρ(r).

equation image(24)

where Uint[ρ] is the classical Coulomb energy, EXC[ρ] is the exchange-correlation energy. Ts[ρ] is the kinetic energy of a fictitious noninteracting system, given by

equation image(25)

The one-electron orbitals {Ψk(r)} are obtained through self-consistent solution of the Kohn–Sham equations

equation image(26)

where veff (r) is the effective potential which is given by

equation image(27)

TD variants of the Hohenberg–Kohn theorems are provided by Deb and Ghosh35 using a steady-state Hamiltonian, equation image containing an external potential vext (r,t) with a periodic temporal behavior. Related TD Kohn–Sham equations are derived35 by minimizing a quasienergy defined as the time average of the expectation value of this Hamiltonian over a period. The TD analog of the Kohn–Sham equation, given by

equation image(28)

to calculate the electron density and the current density using the relations

equation image(29)
equation image(30)

Here, the effective potential depends on the TD density. Some of the functionals depend on current density as well.

Quantum Fluid Dynamics

Madelung obtained the hydrodynamic version32 of the TD Schrödinger equation by considering the polar form of the wave function as,

equation image(31)

where R(r,t) and S(r,t) are real valued functions and R(r,t) ≥ 0.

By equating the real and imaginary parts, we obtain the following equations,

equation image(32)
equation image(33)

The probability density and the current density are, respectively, given as,

equation image(34)

and

equation image(35)

where the velocity of the probability current is equation image.

Equation (32) can now be rewritten as the continuity equation of the probability fluid

equation image(36)

And Eq. (33) can be written as the quantum Hamilton–Jacobi equation (QHJE)36

equation image(37)

In this Euler–Lagrange form, the QHJE resembles the classical Hamilton–Jacobi equation36 apart from the last term which is the quantum potential Vqu. This quantum potential is responsible for all the quantum effects and also the reason behind the quantum nonlocality.

The QHJE in the Lagrangian frame can be written as

equation image(38)

Also the same equation in the Eulerian frame can be partially differentiated with respect to position to yield

equation image(39)

These quantum equations of motion can be solved by computing Vcl and Vqu at different positions using the probability density ρ and the action function S obtained from the continuity equation and the QHJE. Instead of solving the quantum equations of motion, we can make use of the equation of the velocity in terms of the action function.

In contrast to Bohm's analytic approach, the synthetic approach has received much attention amidst the quantum dynamics community starting with the pioneering work of Wyatt and coworkers37 where wave function is constructed by evolving an ensemble of real-valued quantum trajectories. The associated hydrodynamical equations are integrated on the fly using Lagrangian grids. Significant improvement on the numerical scheme was obtained by using arbitrary Euler–Lagrange grids.38, 39 A wide range of problems are solved using this route.38 Quantum trajectory generation in complex space in the formalism of QHJE40, 41 is applied to address issues related to probability density and flux continuity in the complex plane as well as probability conservation along complex quantum trajectories.42–45 For stationary states, QHJE is solved to synthesize the wave function at the same time the reflection and transmission coefficients are calculated for several problems.46, 47 Quantum interference effects are demonstrated within the Bohmian formalism.48, 49Bohm's analytic route is applied to solve problems related to surface science.50, 51 Dynamics of electron nonadiabatic transitions are studied within a quantum trajectory formalism.52, 53 Poirier has developed the bipolar decomposition of wave function in terms of the incident and transmitted parts along with a reflected part. This novel approach is used to study several stationary and scattering problems.54–58 The inherent nonlocality of quantum systems gives rise to phenomena such as entanglement. Quantum entanglement is demonstrated in problems such as wave packet spreading59 and quantum Young's diffraction experiment60 by means of quantum trajectory-based calculation. Bittner and coworkers have used the supersymmetry algebra in solving multidimensional problems involving rare gas clusters with up to 130 atoms.61, 62 The inherent chaotic nature of Bohmian trajectories are explored from the point of view of dynamical systems and the role played by quantum vortices63 in generating chaos is studied. In a similar spirit, a 2D isotropic uncoupled harmonic oscillator model64 is examined and different dynamical features are explained. To bypass instability in quantum trajectory due to singular nature of quantum potential, approximate quantum potential (AQP) calculation has shown promise.65–68 Momentum-dependent quantum potential approach has also been applied along with AQP to determine rate constants of reactions.69

Use of classical dynamics simulation methods to address issues related to quantum phenomena has seen an upsurge of interest.70 Hybrid hydrodynamic-Liouvillian approaches are proposed and applied to solve model problems using both quantum and classical techniques together.71–74 Efforts are made to extend Bohmian mechanics to quantum field theory. Detailed mathematical analysis of such treatment is described in Ref75, 76. A comprehensive study of recent developments has been well-presented in Ref77.

Some Chaotic Anharmonic Oscillators

There has been an extensive study of quantum dynamics of quantum anharmonic oscillator like the Hénon–Heiles oscillator which has been a very well-known system in stellar78 as well as in molecular dynamics.79 The quantum manifestations of classical regular or chaotic dynamics can be well-analyzed by the wave packet dynamics.80–82 Several physical quantities which determine the chaotic behavior of a quantum system can be obtained from the wave packet, Ψ(r,t), at different time steps as a solution to the time dependent Schrödinger equation (TDSE),

equation image(40)

For the Hénon–Heiles system, the potential V has the following form,

equation image(41)

where λ measures the degree of nonlinearity and nonintegrability.83 There are two dimensionless quantities λ and equation image. λ determines the nonlinearity and equation image takes care of the quantum aspects. In the conventional Hénon–Heiles potential, λ = A = B = C = 1 and D = −1. The value of λ can be varied keeping equation image or fix λ = 1 and vary equation image. Hence in the limit equation image, the classical limit can be recovered. λ and equation image cannot be varied simultaneously as the Hénon–Heiles Hamiltonian needs three physical constants: mass, frequency, and length in order that it represents a complete nonlinear system.84 So λ now fixes the length scale as it is inversely proportional to length and no more represents the degree of nonlinearity. Oscillators having the same generalized potential form [Eq. (41)] which have been studied for various purposes include (a) Harmonic oscillator: A = B = 1, λ = 0; (b) Hénon–Heiles oscillator: λ ≠ 0, A = B = C = 1, and D = −1; (c) Barbanis oscillator : λ ≠ 0, A = B = C = 1, and D = 0; (d) Chang-Tabor-Weiss (CTW) oscillator85: λ ≠ 0, A = B = 1, and B = D = 16. Here, the system (a) is linear, (b) and (c) are both nonlinear and nonintegrable and (d) is nonlinear but integrable.

The complete description of a physical system in quantum theory of motion involves both wave and particle aspects. The dynamics is governed by the solution of the TDSE [Eq. (40)]. The motion of a point particle, for a given initial position, steered by this wave is represented by the velocity,

equation image(42)

where S is the phase of the wave function in the polar form [Eq. (31)].

In the Hénon–Heiles system when the classical Hamiltonian is replaced by a properly chosen energy functional from DFT, the QFD equations can be obtained86–89 as the Hamilton's equations of motion,86–91 provided ρ and −S are considered as canonically conjugate variables. The QFD equations for the Hénon–Heiles system can be derived from the following Hamilton's equations.

equation image(43)
equation image(44)

where S1 is −S and the Hamiltonian functional can be expressed as

equation image(45)

The first term in Eq. (45) is the macroscopic kinetic energy, second term is the intrinsic kinetic energy,23 and the last term is the potential energy.

The phase space plots, in terms of two classical dynamic variable which are canonically conjugate, has been used in the extensive study84, 92, 93 of the classical nonlinear systems.

Likewise in the Hénon–Heiles system, the quantum domain behavior is studied in terms of ρ and S1 plots. Numerical solutions86 reveal some interesting features in the phase plots that the system in the classical domain exhibits chaos by showing some fractal-like structures which are absent in the integrable domain. The Shannon entropy, density correlation, and the macroscopic kinetic energy reflect the signatures of chaos showing the effectiveness of the QFD route to quantum chaos.86–88

“An ensemble of particle motions guided by the same wave [Eq. (31)] can be constructed by varying the initial positions in such a way that the probability of finding the particle within this ensemble between r and r + dr at time t is given by ρ(r,t)dr”.36 The so called “Bohmian trajectories” are obtained by the solution of Eq. (42) using different initial positions. It is the concept of trajectories which checks any sensitive dependence on initial conditions and for this study a phase-space distance function has been defined88, 89, 94, 95 as follows,

equation image(46)

where (x, px, y, py) refer to the phase space coordinates of trajectories 1 and 2.

A generalized Lyapunov exponent has also been defined94, 95 to show the variation of the exponential separation between two initially very close trajectories with time.

equation image(47)

The corresponding associated Kolmogorov–Sinai (KS) entropy has also been defined as94, 95

equation image(48)

Quantum chaos can be defined as94, 95 “Quantum dynamics is chaotic, if in a region of phase space the flow of trajectories, according to the Hamilton–Jacobi formulation of quantum mechanics, has positive KS entropy.” These are well-known quantities used to identify chaos as well as regular integrable dynamics depending on the initial conditions in various anharmonic oscillators.

Propagation of a Gaussian wave packet, under the influence of the potential corresponding to a given oscillator, was carried out using the Cayley–Hamilton scheme. The pertinent TDSE [Eq. (40)] is solved to obtain the temporal evolution of Ψ(r,t) using a Peaceman–Rachford-type finite difference algorithm. Once Ψ(r,t) is generated, the trajectories were obtained by solving Eq. (42) at different initial conditions.

Time evolution in distance function D has been shown in Figure 1. Figures 1a–d represent cases a–d, respectively. For Harmonic oscillator (case a), D has a very small value and does not change with time. For classically chaotic oscillators such as the Hénon–Heiles oscillator (case b) and Barbanis oscillator (case c), D oscillates and at times has large values compared to that of the Harmonic oscillator. The CTW oscillator, that is, case d has a small D value which is almost constant with time. Figure 2 shows the associated KS entropies, H for the four systems. For classically integrable cases (a and d), H initially increases and then almost remains constant to a small value. For the classically chaotic cases (b and c), H increases abruptly and has large positive values as compared to the integrable cases. Hence, we can say that two initially nearby trajectories remain close to each other in the integrable case but separate exponentially in the nonintegrable chaotic case. Thus, the quantum theory of motion presents itself as an alternative route to analyze “quantum chaos.”

Figure 1.

Time evolution of phase space distance D for a) harmonic oscillator b) Hénon-Heiles oscillator, c) Barbanis oscillator, and d) CTW oscillator. Figures a) and b) are presented together as is the case with Figures c) and d). (Reprinted with permission from Chattaraj et al. Curr. Sci. 1998, 74, 758, © Indian Academy of Sciences.)

Figure 2.

Time evolution of KS entropy H for a) harmonic oscillator b) Hénon-Heiles oscillator, c) Barbanis oscillator, and d) CTW oscillator. Figures a) and b) are presented together as is the case with Figures c) and d). (Reprinted with permission from Chattaraj et al. Curr. Sci. 1998, 74, 758, © Indian Academy of Sciences.)

The quantum cat map,94, 95 the parabolic barrier,96, 97 hydrogen atom in an oscillating electromagnetic field,98 and also the quantum pinball monitored by measuring devices99 are some examples of the physical systems where quantum chaos has been studied. Konkel and Makowski100 showed that the Bohmian trajectories generated by the combination of two suitably chosen stationary state wave functions generated chaos. Autonomous and nonautonomous flows have been studied by Frisk101 as well as chaos generated by quantum trajectories in zero classical potential. Quantum potential is responsible for the chaotic quantum trajectories in billiards.102

Field-Induced Barrier Penetration in Double-well as well as Single-well potentials

The invariant Kolmogorov–Arnold–Moser or KAM torus103–105 in the phase space characterizes an integrable system in classical mechanics. An introduction of a weak perturbation to the original Hamiltonian does not change the structure of the KAM tori. With increase in the strength of perturbation, the KAM tori disintegrates to give rise to chaos. In the quantum domain behavior, a suppression of chaos along with the stabilization of the dynamics occurs due to the nonclassical features of quantum mechanics106 which has been explained through the study of the quantum domain behavior of double-well and single-well oscillators in an external monochromatic field.107, 108 Quantum theory of motion has been used to generate the Bohmian trajectories in the phase space.

The classical Hamiltonian of a double-well as well as single-well oscillator in presence of an oscillating electric field is given by

equation image(49)

For a given set of parameter values, the trajectories and the stroboscopic plots of the various trajectories can be generated by the solution of the classical Hamilton's equation of motion.109, 110 The phase space may be characterized by stable regions bounded by KAM surfaces or chaotic sea extended over the whole phase space depending on the choice of the initial position and momentum values. The solution of the TDSE provides the TD wave function. Solution of Eq. (42) with different initial positions provides us with the so called “Bohmian trajectories.”

Figure 3 presents the classical stroboscopic plots of the momentum and position variables for the double-well oscillator for two different initial conditions (a) (x = −2.0, px = 0.0) and (b) (x = 2.0, px = 0.0) with the set of parameter values as obtained from the work of Lin and Ballentine.109 The case (a) is in the integrable domain where case (b) gives rise to chaotic dynamics. Figure 4 depicts the corresponding quantal phase space trajectories for the double-well oscillator in presence of an external field with g = 10 and two different initial conditions like its classical counterpart. Here, also we observe regular and chaotic nature of the trajectories corresponding to the two conditions. A regular cantorus-like structure111–113 which is the quantum equivalent to the classical KAM torus is observed. Figure 5 shows the classical stroboscopic plots equation image for a single-well Duffing oscillator in presence of an external field with the initial condition x0 = 0.5, p0 = 0.0 and with (a) g = 1 and (b) g = 30. It is clearly seen that the dynamics of the single-well Duffing oscillator for g = 1 is regular, whereas for g = 30 the system exhibits chaos.

Figure 3.

Classical stroboscopic plot of the momentum and position variables for the double-well oscillator (at t = nT, n = 0,1,2,… …,T = 2π/ω) for two different initial conditions, (x,px) namely, a) (−2.0,0.0) and b) (2.0,0.0) with the set of parameters, (a = −0.5, b = 10.0, g = 10.0 and ω0 = 6.07). (Reprinted with permission from Chattaraj et al. Curr. Sci. 1999, 76, 1371, © Indian Academy of Sciences.)

Figure 4.

Quantal phase space trajectories for the double-well oscillator for two different initial conditions, (x,px) namely, a) (−2.0,0.0) and b) (2.0,0.0) with the set of parameters, (a = −0.5, b = 10.0, g = 10.0 and ω0 = 6.07). (Reprinted with permission from Chattaraj et al. Curr. Sci. 1999, 76, 1371, © Indian Academy of Sciences.)

Figure 5.

Classical stroboscopic plots equation image for a single-well Duffing oscillator in presence of an external field with the initial condition: x0 = 0.5, p0 = 0.0 and with a) g = 1 and b) g = 30. (Reprinted with permission from Sengupta et al. Ind. J. Chem. A, Theor. Chem. (Spl. Issue) 2000, 39, 316, © NISCAIR PUBLICATIONS.)

For the investigation of the breakdown of KAM torus (case a) along with the quantal suppression of classical chaos, the behavior of the double-well oscillator in presence of an external field with two other amplitudes, g = 20 and g = 40 with the same initial conditions as in case a with g = 10 was studied. Figures 6 and 7 depict the time evolution of KS entropies corresponding to the classical motion as well as quantum trajectories associated with the double-well oscillator. It is clearly observed that for g = 20 the classical system exhibits chaos, whereas the quantum trajectories behave regularly similar to the situation for g = 10, clearly pointing out the fact of quantum suppression of classical chaos in the system. The plots of the time evolution of the associated KS entropies for the classical motion of the single-well Duffing oscillator in the two cases (g = 1 and g = 30) are presented in Figure 8 and their corresponding quantal plots are presented in Figure 9. From the figures, it can be concluded that the classical as well as the quantal regular motion provide small H values compared to the corresponding chaotic motion.

Figure 6.

Plot of KS entropy (H) as a function of time (t) associated with the classical motion for the double-well oscillator in presence of external field with g = 10, g = 20 and g = 40 with initial condition (x0 = −2.0, pmath image = 0.0). (Reprinted with permission from Chattaraj et al. Curr. Sci. 1999, 76, 1371, © Indian Academy of Sciences.)

Figure 7.

Plot of KS entropy (H) as a function of time (t) associated with the quantal motion for the double-well oscillator in presence of external field with g = 10, g = 20 and g = 40 with initial condition (x0 = −2.0, pmath image = 0.0). (Reprinted with permission from Chattaraj et al. Curr. Sci. 1999, 76, 1371, © Indian Academy of Sciences.)

Figure 8.

Time evolution of the KS entropy associated with the classical motion of the single-well Duffing oscillator in presence of an external field with the initial condition: x0 = 0.5, p0 = 0.0 and with a) g = 1 and b) g = 30. (Reprinted with permission from Sengupta et al. Ind. J. Chem. A, Theor. Chem. (Spl. Issue) 2000, 39, 316, © NISCAIR PUBLICATIONS.)

Figure 9.

Time evolution of the KS entropy associated with the quantal motion of the single-well Duffing oscillator in presence of an external field with the initial condition: x0 = 0.5, p0 = 0.0 and with a) g = 1 and b) g = 30. (Reprinted with permission from Sengupta et al. Ind. J. Chem. A, Theor. Chem. (Spl. Issue) 2000, 39, 316, © NISCAIR PUBLICATIONS.)

The phase volume can be defined as110, 114, 115

equation image(50)

Figure 10 shows the time evolution of the phase volumes at different values of g associated with the quantal motion of the double-well oscillator. It reflects the same behavior as studied by the KS entropies. Figure 11 shows the time evolution of phase volume of the single-well Duffing oscillator for the two cases (g = 1 and g = 30). The aforementioned suppression of chaos was observed116 in an N-component ∅4- oscillator in presence of an external field. The observations of Konkel and Makowski,100 Frisk,101 and Sales and Florencio102 are somewhat opposite to that of the quantum suppression of classical chaos. They have, however, termed such effects as “classical suppression of Bohmian Chaos.”

Figure 10.

Plot of phase volume (Vps) as a function of time (t) associated with the quantal motion for the double-well oscillator in presence of external field with g = 10, g = 20 and g = 40 with initial condition (x0 = −2.0, pmath image = 0.0). (Reprinted with permission from Chattaraj et al. Curr. Sci. 1999, 76, 1371, © Indian Academy of Sciences.)

Figure 11.

Time evolution of the phase volume for the single-well Duffing oscillator in presence of an external field with the initial condition: x0 = −0.5, p0 = 0.0 and with a) g = 1 and b) g = 30. (Reprinted with permission from Sengupta et al. Ind. J. Chem. A, Theor. Chem. (Spl. Issue) 2000, 39, 316, © NISCAIR PUBLICATIONS.)

KAM Transition

With the variation of the nonintegrability parameter, many classical nonlinear dynamical systems change from a regular to a chaotic one by exhibiting the KAM transition.78, 84, 92, 93, 117 The quantum mechanical analog of the KAM transition has been a subject matter of research118, 119 for quite some time. For this purpose, the classical and quantum Hénon–Heiles oscillators have been extensively studied by our group.120 Systems like coupled nonlinear oscillators,78 kicked rotators,121 and so forth have been studied to understand their chaotic behaviors in the classical domain. Initially, the classical Hénon–Heiles oscillator in the regular domain is considered and then exposed it to an external field. With an increase in the intensity of the field, the oscillator dynamics becomes chaotic from regular as has been noticed by Lin and Ballentine109 for the field-induced barrier penetration in a double-well oscillator and its quantum analog.107, 122–126

The classical Hamiltonian of the Hénon–Heiles oscillator in presence of an external axial field applied in the y-direction is expressed as,

equation image(51)

The first term is the kinetic energy, second and third terms are the harmonic and anharmonic parts of the potential energy of the oscillator, whereas the last term represents the interaction of the oscillator with the external axial field. The TDSE has been solved to obtain Ψ(r,t) and the Eq. (42) is solved to obtain the Bohmian trajectories.

Figure 12 presents the classical and quantal phase-portraits (y,py) for four different field intensities (g = 0.0,0.1,0.5,1.0). At g = 0.0, the classical case shows a distinct torus and the quantum analog shows a cantorus-like structure. With increase in the field intensity, the system exhibits a typical KAM-type transition to the chaotic domain. This behavior is also reflected in the plots of time evolution of distance function and Kolmogorov-Sinai-Lyapunov entropy as shown in Figures 12, 13, and 14. Quantum domain behavior of the oscillators possesses a direct link with the nature of their classical variants. The zero quantum potential limit of these systems is being explored.

Figure 12.

Classical and quantal phase portraits (py vs. y) for the Henon–Heiles oscillator in presence of an external field of varying intensities with g = 0.0, 0.1, 0.5, and 1.0. Also shown are the respective phase space distance functions (D(t)). (Reprinted with permission from Chattaraj et al. J. Chem. Sci (Special Issue on 10th CRSI National Symposium, Invited Article) 2008, 120, 33, © Springer.)

Figure 13.

Time evolution of the KSL entropy associated with the classical motion (HCl) of the Henon–Heiles oscillator in the presence of an external field of varying intensities with g = 0.0, 0.1, 0.5, and 1.0. (Reprinted with permission from Chattaraj et al. J. Chem. Sci (Special Issue on 10th CRSI National Symposium, Invited Article) 2008, 120, 33, © Springer.)

Figure 14.

Time evolution of the KSL entropy associated with the quantal motion (HQu) of the Henon–Heiles oscillator in the presence of an external field of varying intensities with g = 0.0, 0.1, 0.5, and 1.0. (Reprinted with permission from Chattaraj et al. J. Chem. Sci (Special Issue on 10th CRSI National Symposium, Invited Article) 2008, 120, 33, © Springer.)

Rydberg Atoms

The chaotic dynamics of a quantum system such as the Rydberg atom in an oscillating electric field has been an area of interest in both theoretical and experimental studies. These Rydberg atoms have been the genuine sources for investigating the quantum aspects of chaos.127 Hydrogen atom, when placed in an external oscillating field, exhibits regular to chaotic transition depending on the frequency and field intensity.128 Hydrogen atom is one of the simplest solvable quantum mechanical systems. A Keplerian system, which is known to be the classical mechanical counterpart of the hydrogen atom, is known to exhibit chaotic motion in presence of external field. For this reason, our group122–126 has tried to look for the quantum signature of chaos in an electronically excited hydrogen atom in presence of an external field. To understand the chaotic behavior associated with quadratic Zeeman effect, extensive theoretical129–131 and experimental132, 133 studies have been carried out regarding this system.

The time evolution of the ground state (n = 1) and excited state (n = 20) of a hydrogen atom subjected to an oscillating electric field has been analyzed in the quantum trajectory study by Chattaraj and Sengupta.122 Figures 15 and 16 show the plots of Shannon entropy and density correlation function. In both the figures, a and b refer to the ground state and n = 20 state of the hydrogen atom, respectively. From the figures, it is clear that the applied field drastically changes the dynamics of Ψ20 state of H-atom unlike the ground state. Figures 15 and 16 show that for n = 1 state the entropy and correlation values do not change on the application of external field. But show significant changes for n = 20 case. The temporal evolution of phase space distance function of two initially close trajectories and the associated KS entropy clearly distinguishes the chaotic nature of n = 20 state as compared to the integrable n = 1 state in the presence of the external field.

Figure 15.

Time evolution of S/k, where S is the Shannon entropy and k is the Boltzman constant, for Rydberg atom in external field: a) n = 1 b) n = 20. (Reprinted with permission from Chattaraj and Sengupta Curr. Sci. 1996, 71, 134, © Indian Academy of Sciences.)

Figure 16.

Time evolution of density correlation function C for Rydberg atom in external field: a) n = 1, b) n = 20. (Reprinted with permission from Chattaraj and Sengupta Curr. Sci. 1996, 71, 134, © Indian Academy of Sciences.)

In principle, the propagation of quantum trajectories needs the exact solution of Schrödinger equation for both stationary and nonstationary states. The information regarding these trajectories helps in direct computation of the observables in terms of probability amplitude and phase of the wave function. Madelung, de Bröglie, Bohm, Takabayashi, and others developed the fluid dynamical formulation of quantum mechanics through the dynamics of related quantum trajectories. The de Bröglie–Bohm interpretation of quantum mechanics is related to the hydrodynamic formulation, the foundations of which were build up within 1926–1954. New methods for computing quantum trajectories were introduced in the late 90s and since then strong and effective computational methods have been developed which helped in the solution of diverse range of problems. Depending on the procedures of computation, the investigations that involved quantum trajectories were divided into two broad categories. First one is the de-Broglie Bohm analytic approach which commences with the solution of the TDSE by conventional computational techniques and then the individual “particles” are evolved along quantum trajectories with velocities that has been generated by the “???-field”. The second category is known as the synthetic approach where computation of the trajectories and the hydrodynamic fields are taken care of with an “on the fly” approach. A detailed discussion on such quantum trajectory methods can be found in Ref38.

Quantum Hydrodynamics in Many-particle Systems

Although QFD of Madelung was essentially for a single particle, it is extended to many-particle systems within Hartree theory,134, 135 Hartree–Fock theory,136, 137 and natural orbitals theory.138 According to the TDDFT, a TD quantity can be expressed as a functional of the TD probability density and current density ρ(r,t) and j(r,t), respectively. QFDFT8–10, 11–16 is the amalgamation of TDDFT and QFD, and it has been shown that the latter two are formally equivalent.5, 7 For the many-particle systems, the system dynamics is governed by the generalized nonlinear Schrödinger equation (GNLSE)8–10, 11–16 given by,

equation image(52)

The N-particle dynamics is simulated through the dynamics of N-independent particles under the influence of an effective potential Veff (r,t). This one-body potential depends on the Hohenberg–Kohn functional22, 26 and is nonlinear in nature as in Hartree–Fock or Kohn–Sham theories. The expression for the probability density is given by

equation image(53)

and for current density as,

equation image(54)

Thus, the dynamics of any system can be obtained from ρ(r,t) and j(r,t). QFDFT has been successfully applied in the study of various TD processes.

Time Evolution of Chemical Reactions

TD processes like collisions between atoms and molecules with protons have been a subject of interest in recent years.139–141 Such collision processes are very important in nuclear physics, astrophysics as well as in chemical reactions. Ion-atom and ion-molecule reactions play an important role in various chemical reactions.142 These collision processes provide insights into the famous hard-soft-acid-base (HSAB) principle21, 143, 144 given by Pearson. Pearson also proposed another related hardness principle known as the maximum hardness principle (MHP).143, 145–149 In various chemical and physical processes, the HSAB principle demands validity of the MHP.150–153 The inverse relationship154–157 between hardness and polarizability gives rise to the minimum polarizability principle (MPP)16, 158 which states that “the natural direction of evolution of any system is toward a state of minimum polarizability.” DFT21, 22, 26 has successfully calculated hardness and polarizability of different chemical reactions. Two very important aspects of DFT, namely, the TDDFT5, 7 and the excited state DFT159 have been used to analyze the HSAB principle in the light of MHP and MPP. There is no general excited-state DFT except for systems involving states of lowest energy for a given symmetry class1, 2, 160 and two-state and multistate ensembles.161–165 Time evolution of various reactivity parameters during ion-atom34, 166, 167 and ion-molecule168 processes have been studied in detail by our group.

Hardness (η)169, 170 for an N electron system with energy E can be defined as,

equation image(55)

where μ and v(r) are chemical potential (Lagrange multiplier associated with normalization constraint of DFT) and external potential, respectively. Hardness can also be defined as171, 172

equation image(56)

where f(r) is the Fukui function173 and η(r,r) is the hardness kernel given, in terms of the Hohenberg–Kohn–Sham universal functional of DFT,22, 26 F[ρ], as

equation image(57)

According to the HSAB principle, proton being a hard acid prefers to bind to the systems in their ground state where they are the hardest. With excitations or with increasing contribution of the excited state in a two-state ensemble, this preference keeps on decreasing. This process is verified using TDDFT and excited state DFT. The solution of the GNLSE within a quantum fluid density functional framework8, 16 provides us with the dynamic profiles of various reactivity parameters associated with the ion-atom and ion-molecule collision processes. Ref16 provides the numerical details. For the two state ensemble case, the density is represented as,

equation image(58)

where ρgs and ρes are ground state174 and excited state175 densities, respectively, and ω is a real number.161–165, 176–179 The TD polarizability can be calculated as follows,

equation image(59)

where Dmath image (t) is the electronic part of the induced dipole moment and Gz(t) is the z-component of the external Coulomb field.

The collision process can be divided into three distinctive zones, namely, approach, encounter, and departure. Figures 17 and 18 portray the dynamic profiles of hardness η and polarizability α for ground state (1S) and various excited states (1P, 1D, 1F) of He isoelectronic atoms/ions colliding with a proton. It is clear that in the encounter region of the collision process the hardness gets maximized and the polarizability gets minimized clearly indicating a favorable dynamical process. In the encounter region, the electron density is shared by both the nuclei as a result hardness has large values. The central nature of the Coulomb potential originating from a single nucleus reflects the spherical nature of the electron cloud in the isolated atom/ion. With electronic excitation, any system becomes softer and more polarizable.180, 181 Hence, η1S > η1P > η1D > η1D and α1S < α1P < α1D < α1F for any given species, which also prove the validity of MHP and MPP. As proton is a hard acid, it prefers to bind with X (X= He, Li+, Be2+, B3+, C4+) in the order 1S > 1P > 1D > 1F. Hence, the validity of the HSAB principle is provided166 in a dynamical context as the maximum η value decreases and the minimum α value increases in the order 1S → 1P → 1D → 1F. Figures 19 and 20, respectively, show the time evolution of hardness and polarizability of noble gas systems namely, He, Ne, Ar, Kr, Xe during protonation. As expected, the hardness and polarizability attain, respectively, the maximum and the minimum in the encounter region. As we proceed from He to Xe, the maximum η value decreases and the minimum α values increases. Hence, Xe becomes the most reactive among the lot which supplements the fact that first compound of noble gas elements was of Xe.

Figure 17.

Time evolution of hardness (η, au) during a collision process between an X - atom/ion (X) He, Li+, Be2+, B3+, C4+) in various electronic states (1S, 1P, 1D, 1F) and a proton. (Reprinted with permission from Chattaraj and Maiti J. Am. Chem. Soc. 2003, 125, 2705, © American Chemical Society.)

Figure 18.

Time evolution of polarizability (α, au) during a collision process between an X-atom/ion and a proton. See the caption of Figure 17 for details. (Reprinted with permission from Chattaraj and Maiti J. Am. Chem. Soc. 2003, 125, 2705, © American Chemical Society.)

Figure 19.

Time variation of hardness (η, au) of He, Ne, Ar, Kr, Xe during protonation. (Reprinted with permission from Chattaraj and Maiti J. Am. Chem. Soc. 2003, 125, 2705, © American Chemical Society.)

Figure 20.

Time variation of polarizability (α, au) of He, Ne, Ar, Kr, Xe during protonation. (Reprinted with permission from Chattaraj and Maiti J. Am. Chem. Soc. 2003, 125, 2705, © American Chemical Society.)

Reactants collide with each other to result in product formation in a chemical reaction. The ion-molecule reactions play an important role to explain molecule formation in dense interstellar clouds182 as well as in various chemical systems.142 The presence of hydrogen in the interstellar medium results in reactions involving proton transfer. As the neutral species are able to bind proton, all atoms and molecules can be considered to be bases. These chemical reactions can be well-understood by the study of various reactivity parameters such as hardness and polarizability associated with a collision process. Figures 21 and 22 provide us with the time evolution of hardness and polarizability during the protonation of N2 molecule in ground and excited states. During protonation, in the encounter regime, hardness is maximized and polarizability is minimized and they are symmetric in both ground state and excited electronic state of N2 molecule which can be observed from the figures. Owing to the Coulomb singularity, hardness becomes exceptionally large at the point of closest approach of the two nuclei. The first excited electronic state of N2, being softer and more polarizable, has lower ηmax and higher αmin values than that of the ground state N2. Hence, excited state N2 is more reactive than ground state N2 molecule. Figures 23 and 24 show the time evolution of hardness and polarizability during the protonation of HF molecule both in ground and excited states. The regioselectivity in a reaction involving a multiple-site molecule in both its ground and first-excited electronic states is shown by the reactions involving heteronuclear molecules. In HF molecule, the electron densities are centered on H and F nuclei, and the Mulliken charges of H and F are 0.5146 and 9.4854, respectively. The figures show the TD hardness and polarizability profiles for protonation considering attack on both sides of HF. According to the figures as well as from MHP and MPP, the F-site in HF is kinetically more favorable to protonation in both the electronic states. As a result, proton attacks the harder F-site to form the more stable H2F+ cation. Protonation is less favorable in the excited state than the ground state as ηmax gets smaller and αmin gets larger during excitation. With the help of HSAB principle, the chemical process as well as regioselectivity can be analyzed better. The QFDFT formalism helps us in understanding the dynamical behavior of chemical reaction involving proton and an atom or a molecule. QFDFT also explains regioselectivity very successfully.

Figure 21.

(left) Time (a.u.) evolution of hardness (η, a.u.) during a collision process between a nitrogen molecule in its ground state and a proton. (right) Time (a.u.) evolution of hardness (η, a.u.) during a collision process between a nitrogen molecule in its first excited state and a proton. (Reprinted with permission from Chattaraj and Maiti J. Phys. Chem. A 2004, 108, 658, © American Chemical Society.)

Figure 22.

(left) Time (a.u.) evolution of polarizability (α, a.u.) during a collision process between a nitrogen molecule in its ground state and a proton. (right) Time (a.u.) evolution of polarizability (α, a.u.) during a collision process between a nitrogen molecule in its first excited state and a proton. (Reprinted with permission from Chattaraj and Maiti J. Phys. Chem. A 2004, 108, 658,© American Chemical Society.)

Figure 23.

(left) TD hardness (η, a.u.) profile for the protonation of HF in its ground state considering the attack from both the hydrogen and fluorine sides. (right) TD hardness (η, a.u.) profile for the protonation of HF in its first excited state considering the attack from both the hydrogen and fluorine sides. (Reprinted with permission from Chattaraj and Maiti J. Phys. Chem. A 2004, 108, 658, © American Chemical Society.)

Figure 24.

(left) TD polarizability (α, a.u.) profile for the protonation of HF in its ground state considering the attack from both the hydrogen and fluorine sides. (right) TD polarizability (α, a.u.) profile for the protonation of HF in its first excited state considering the attack from both the hydrogen and fluorine sides. (Reprinted with permission from Chattaraj and Maiti J. Phys. Chem. A 2004, 108, 658, © American Chemical Society.)

Reactivity Dynamics of Atoms in Ground and Excited States

Behavior of highly excited atoms in presence of an external field is an important area of research.183–188 Study of the Rydberg atom has seen an upsurge of interest.183 Depending on the amplitude and frequency of external field, these Rydberg atoms exhibit interesting features. Experimentalists showed184–186 that ionization of the atom takes place only when threshold field amplitude or field strength is reached. The ionization of hydrogen atom increases monotonically and drastically once the threshold is obtained. With increase in the field amplitude, the ionization continues until it is fully ionized. The ionization of helium takes place through a series of steps with stable regions in between. Chaotic ionization of Rydberg helium atom is observed187, 188 at a lower field than that required for the Rydberg hydrogen atom. But for complete ionization of helium, the field required is more. This difference is mainly due to the non-Coulombic core potential of helium.

For a system of N-electrons and total energy E, electronegativity can be defined as,

equation image(60)

where μ and v(r) are the chemical potential and external potential, respectively. Hardness increases when a system goes from a regular to a chaotic region.189

The knowledge about N and v(r) is necessary to illustrate an N-particle system completely under the influence of an external potential v(r). χ and η of a system are given by changing N at a given v(r) and α is the response by changing v(r) at a fixed N. The study of the time evolution of various reactivity parameters for different atoms in their ground states and excited states in presence of external field of varying intensities and frequencies has been done by our group.34, 126, 190 The pertinent TDSE has been solved for the hydrogen ground and excited states and helium excited state, whereas solution of the GNLSE provides us with the overall dynamics of the ground state of helium atom. The TDSE is written as,

equation image(61)

where v1core (r) = −(1/r) for ground and excited states of hydrogen atom and = vn (r) + vs (r) + vp (r) for excited state of He atom. vn (r) is the Coulomb potential between the unshielded helium nucleus and the Rydberg electron, which is given as −(2/r). vs (r) is the potential187, 188 due to shielding and vp (r) is the core polarization. The explicit forms of vs (r) vp (r), and vext(r) can be found in Ref126. For helium, ground state GNLSE is solved. The effective potential in the GNLSE can be found in Ref126.

A TD energy functional has been defined5, 7, 32, 35, 191 as

equation image(62)

where vcore(r) = v1core(r) = −(1/r) for hydrogen ground state and excited state and helium excited state and vcore(r) = v2core(r) = −(2/r) for helium ground state.

The chemical potential is given as,

equation image(63)

and electrophilicity index (ω) is given as192–195

equation image(64)

It measures the propensity to absorb electrons.

In the figures GS and ES, respectively, refer to the ground and excited electronic states of the systems and two different intensities, viz. 3.509 × 1012 and 3.509 × 1016 W/cm2, respectively, for hydrogen and helium are used. Chemical hardness (η) is higher in the ground state, for both hydrogen and helium atoms, than that of the excited state for both colors and intensities of the external field. This is clear from Figures 25 and 26. In ground state, ηHe > ηH but in excited state ηH > ηHe, reflecting the enhanced chaotic behavior in helium-excited state which in turn clarifies the lower threshold amplitude for helium. Figures 27 and 28 represent the time dependence of electrophilicity index (ω) for H and He, respectively. Opposite to the behavior of chemical hardness, ωGS < ωES, for both hydrogen and helium. Both the ground state hydrogen and helium atoms are harder and less polarizable characterized with smaller phase volume, chemical potential, and electrophilicity index than their excited state counterparts.

Figure 25.

Time evolution of chemical hardness (η, a.u.) when a hydrogen atom is subjected to external electric fields. (GS, ground state; ES, excited state): (–), monochromatic pulse; (–⊙–), bichromatic pulse. Maximum amplitudes (ε0) = 0.01 a.u. (intensity = 3.509 × 1012 W/cm2); 1 a.u. (intensity = 3.509 × 1016 W/cm2), t′ = (5π/ω0), ω0 = π, ω1= 2ω0. (Reprinted with permission from Chattaraj and Sarkar Int. J. Quantum Chem. 2003, 91, 633, © Wiley Periodicals, Inc..)

Figure 26.

Time evolution of chemical hardness (η, a.u.) when a helium atom is subjected to external electric fields. (GS, ground state; ES, excited state): (–), monochromatic pulse; (–⊙–), bichromatic pulse. Maximum amplitudes (ε0) = 0.01 a.u. (intensity = 3.509 × 1012 W/cm2); 1 a.u. (intensity = 3.509 × 1016 W/cm2), t′ = (5π/ω0), ω0 = π, ω1= 2ω0. (Reprinted with permission from Chattaraj and Sarkar Int. J. Quantum Chem. 2003, 91, 633, © Wiley Periodicals, Inc.)

Figure 27.

Time evolution of electrophilicity index (ω, a.u.) when a hydrogen atom is subjected to external electric fields. See the figure caption of Figure 25 for details. (Reprinted with permission from Chattaraj and Sarkar Int. J. Quantum Chem. 2003, 91, 633, © Wiley Periodicals, Inc.)

Figure 28.

Time evolution of electrophilicity index (ω, a.u.) when a helium atom is subjected to external electric fields. See the figure caption of Figure 26 for details. (Reprinted with permission from Chattaraj and Sarkar Int. J. Quantum Chem. 2003, 91, 633, © Wiley Periodicals, Inc..)

The generation of the higher-order harmonics196–198 during TD processes like atom-field interaction are areas of much interest in modern research. The harmonic spectrum is generated by calculating the Fourier transform of the induced dipole moment appearing in Eq. (59) to obtain d(ω). The absolute square of the Fourier transform |d(ω)|2 has been shown199 to be roughly proportional to the experimental harmonic distribution. Figure 29 presents the harmonic spectra of He atom in ground state (1s) and excited state (1s2p) when the field amplitude ε0 = 0.01 a.u., for both monochromatic (frequency ω0 = π) and bichromatic (frequency ω1 = 2 ω0) laser pulses. The envelopes of the plots are similar to that reported in Ref.200.

Figure 29.

Harmonic spectra of He atom in ground state and excited state subjected to external electric fields: (−) monochromatic pulse; (···) bichromatic pulse. Maximum amplitudes (ε0) = 0.01au. ω0 = π; ω1 = 2ω0. (Reprinted with permission from Chattaraj and Maiti J. Phys. Chem. A 2001, 105, 169, © American Chemical Society.)

Effect of Confinement on Chemical Reactivity

Confined many electron systems have been an interesting area of research due to their wide applications such as atom/molecules in cavities, impurities, and excitons in semiconductor nanostructures, quantum wells, and quantum dots.201 The physical properties of confined atoms as well as confined molecules depend on the confinement volume.202–207 There have been wide applications of confined quantum systems such as in physics, chemistry, and biology.201, 208–212 The models of confined quantum systems such as confined atoms and confined molecules have been used to study matters under high pressure. The confined systems and their free counterpart behave differently owing to the fact that confinement brings about modifications of certain properties.201, 208–215. A detailed study about the effect of confinement, both spherical213–215 as well as cylindrical,216 on the chemical reactivity of atoms have been pursued. In these studies, the confinement is incorporated through the Dirichlet boundary condition, that is, a vanishing wave function on the surface of the confining box has been considered.

Figures 30 and 31 show the variation of softness and polarizability with cutoff radius of the spherical box, respectively. It is clear that the systems become harder and less polarizable with decrease in the cutoff radius of the spherical box. At the same time, it is more difficult to have excitation. Figure 32 shows the change in polarizability with the change of cutoff radius for ions confined in a spherical box. It is clear that on ionization, a system becomes less polarizable. The inverse relationship between softness (S) and α1/3 has been shown in Figure 33 for ions. The regression coefficient values have been given alongside the individual curves. It has been shown by many researchers144, 155–157 for atoms, molecules, and clusters that the behavior of S as a function of (α1/3) has been linear. Figures 34 and 35 present the plots of scaled hardness against the cutoff radius for atoms and ions, respectively. It is clear that with decrease in cutoff radius of the spherical box ηs increases. Fluorine has the highest ηs value and lithium has the lowest, whereas C4+ has the highest and C+ the lowest value as seen from Figures 34 and 35. Here, the scaled hardness has been defined as,

equation image(65)

where η is the hardness and μ is the chemical potential. The scaled hardness is a better quantity to locate fixed points in the hardness profile as opined by Ghanty et al.217

Figure 30.

Plot of softness (S, a.u.) versus cutoff radius (R, a.u.) for atoms confined in a spherical box. (Reprinted with permission from Chattaraj and Sarkar Chem. Phys. Lett. 2003, 372, 805, © Elsevier.)

Figure 31.

Plot of polarizability (α, a.u.) versus cutoff radius (R, a.u.) for atoms confined in a spherical box. (Reprinted with permission from Chattaraj and Sarkar Chem. Phys. Lett. 2003, 372, 805, © Elsevier.)

Figure 32.

Plot of polarizability (α, au) versus cutoff radius (R, au) for ions confined in a spherical box. (Reprinted with permission from Chattaraj and Sarkar J. Phys. Chem. A 2003, 107, 4877, © American Chemical Society.)

Figure 33.

Plot of softness (S, a.u.) versus α1/3 (a.u.) for ions confined in a spherical box. (Reprinted with permission from Chattaraj and Sarkar J. Phys. Chem. A 2003, 107, 4877, © American Chemical Society.)

Figure 34.

Plot of scaled hardness (ηs, a.u.) versus cutoff radius (R, a.u.) for atoms confined in a spherical box. (Reprinted with permission from Chattaraj and Sarkar J. Phys. Chem. A 2003, 107, 4877, © American Chemical Society.)

Figure 35.

Plot of scaled hardness (ηs, au) versus cutoff radius (R, au) for ions confined in a spherical box. (Reprinted with permission from Chattaraj and Sarkar J. Phys. Chem. A 2003, 107, 4877, © American Chemical Society.)

The study of the dynamic profiles of various reactivity parameters during TD processes such as ion-atom collisions and atom-field interactions within a confined cylindrical box, have been studied in detail.216 In this study, we have considered cylindrical polar coordinates equation image. The confinement is incorporated through the Dirichlet-type boundary condition. The effect of confinement is brought about by varying the length of the cylinder keeping the radius constant. The solution of the GNLSE provides us with the TD density and current density which in turn give us the time evolution of various reactivity parameters such as hardness, polarizability, electrophilicity, and so forth. The aforementioned work deals with the response of the system under simultaneous effects of confinement and excitation. Figure 36 shows the dynamic profile of hardness when a helium atom in ground state (GS) and excited state (ES) is placed in an intense laser field of amplitude (ε0) 10-6, 0.01 and 100 a.u. For ground state the unconfined system is represented by length of cylinder = 6 a.u. The length of cylinder = 4.8 a.u. and 4.2 a.u. represent the two confined systems respectively. For the excited state the unconfined system is represented with length of cylinder = 7.2 a.u. The length of cylinder = 6.6 a.u. and 5.8 a.u. represent the two confined systems respectively. The frequency (ω0) of the field is equal to π. From the figure, it is clear that the excited state hardness is lower than that of the ground state hardness both in free as well in confined systems, a clear cut signature of MHP. With increase in degree of confinement, the hardness decreases both in ground and excited states. Hence, from the figure, it is clear that by exciting and increasing the degree of confinement the ground state hardness profile can be retrieved. Figure 37 represents the time evolution of hardness and polarizability associated with the collision process between a proton and helium atom in its ground state (1S) and excited state (1P) within a confined environment. The hardness attains a maximum value and the polarizability attains a minimum value in the encounter region. It is clear that with an increase in the amount of confinement the maximum η value increases and minimum α value decreases. Hence, from MHP and MPP, we can say that the system tends to get stability as the degree of confinement increases. According to HSAB principle, proton being a hard acid prefers to bind with the most confined helium atom in ground as well as in excited states. Increase in maximum η value and decrease in minimum α value with increase in confinement proves the validity of the HSAB principle in a dynamical context. The dynamic profiles of various other reactivity indices such as the chemical potential, electrophilicity, polarizability, and entropy in presence of varying electric fields under confinement have also been studied.216 From these profiles, along with the hardness profile, it is clear that the stability of the system increases with an increase in the degree of confinement. Simultaneous excitation and confinement of a system may make the system equivalent to the corresponding free system in the ground state and the dynamics associated with the TD processes exhibit expected trends.

Figure 36.

Time evolution of hardness (η, a.u.) when a helium atom in the ground state (G.S.) and the excited state (E.S.) is placed in an intense laser field (amplitude = 10−6, 0.01 and 100 a.u.). For the ground state, length of the cylinder= 6.0 a.u. represents the unconfined system, length of the cylinder = 4.8 a.u. and length of the cylinder = 4.2 a.u. represent confined systems. For the excited state, length of the cylinder = 7.2 a.u. represents the unconfined system, length of the cylinder = 6.6 a.u. and length of the cylinder = 5.8 a.u. represent confined systems (radius of the cylinder = 13.220496 a.u.). ω0 = π. (Reprinted with permission from U. Sarkar, M. Khatua and P. K. Chattaraj, Phys. Chem. Chem. Phys. 2012, 14, 1716, © RSC Publishing.)

Figure 37.

Time evolution of hardness (η, a.u.) and polarizability (α, a.u.) during a collision process between a proton and helium atom in ground state and excited state. See the figure caption of Figure 36 for details. (Reprinted with permission from U. Sarkar, M. Khatua and P. K. Chattaraj, Phys. Chem. Chem. Phys. 2012, 14, 1716, © RSC Publishing.)

Study of confined molecules has also been an area of interest. There have been various theoretical218, 219 as well as experimental220, 221 studies regarding confined molecules. Inertially confined molecular ions interacting with strong laser pulses can be a very efficient source of high-order harmonics.222 Orientation dependence of the ionization of CO and NO has been studied in an intense femtosecond two-color laser field.223 Quantum translation-rotation dynamics of confined hydrogen molecules in small dodecahedral cage of a clathrate hydrate has been studied224 as well as standard density functional calculations regarding structure, stability, and reactivity of clathrate hydrates with hydrogen encapsulation have also been carried out.225 Study of the reactivity dynamics of diatomic molecules in ground and excited electronic states during various TD processes has been done within a quantum fluid density functional framework.226 In both the processes (collision as well as molecule–field interaction) with an increase in the degree of confinement, the system attains stability in both the electronic states as dictated by the dynamical variants of the associated electronic structure principles.

Conclusions

Various quantum potential-based approaches such as QFD, quantum theory of motion, and QFDFT have been shown to be useful in analyzing the quantum domain behavior of classical nonintegrable systems. Important insights into different TD quantum mechanical processes such as ion-atom and ion-molecule collisions, atom–field and molecule–field interactions, effects of electronic excitation and confinement therein, and so forth have been obtained through the associated density dynamics.

Acknowledgements

The authors thank Dr. Madhusudan Singh for the invitation to write this review article. Authors also thank Drs. S. Sengupta, A. Poddar, B. Maiti, and U. Sarkar for their help in various ways.

Biographical Information

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Pratim Kumar Chattaraj obtained his B.Sc. and M.Sc. degrees in Chemistry from Burdwan University and his Ph.D. degree from Indian Institute of Technology Bombay. Presently, he is a professor in the Department of Chemistry, Indian Institute of Technology Kharagpur and also the Convener of the Center for Theoretical Studies there. He was a postdoctoral research associate in the University of North Carolina (Chapel Hill). His research interests include density functional theory, the theory of chemical reactivity, aromaticity in metal clusters, ab initio calculations, quantum trajectories, and nonlinear dynamics. He has been invited to deliver special lectures at several international conferences and to contribute chapters to many edited volumes. Professor Chattaraj is a member of the editorial board of J. Mol. Struct. (Theochem), J. Chem. Sci., and Ind. J. Chem. A among others. He was the Head of the Department and a council member of the Chemical Research Society of India. He is a Fellow of the Indian Academy of Sciences (Bangalore), the Indian National Science Academy (New Delhi), the National Academy of Sciences, India (Allahabad), and the West Bengal Academy of Science and Technology. He is a J. C. Bose National Fellow. He has edited several books, Chemical reactivity theory: A density functional view, Aromaticity and metal clusters, and Quantum trajectories, published by Taylor and Francis Books, Inc./CRC Press, Boca Raton, FL, and a special issue each of J. Chem. Sci. (on “Chemical Reactivity”) and J. Mol. Struct. (Theochem) (with Professor A. J. Thakkar, on “Conceptual aspects of electron densities and density functionals”). He is the Editor of the book series on “Atoms, Molecules, and Clusters: Structure, Reactivity, and Dynamics” and of a two volume set (with Professor S. K. Ghosh) on “Concepts and Methods in Modern Theoretical Chemistry, Volume I: Electronic Structure and Reactivity; Volume II: Statistical Mechanics” both published by Taylor and Francis Books.

Biographical Information

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Munmun Khatua received her B.Sc. (Chemistry Honors) and M.Sc. (Chemistry) degrees from the Vidyasagar University, West Bengal. She joined the research group of Professor P. K. Chattaraj for her Ph.D. degree in the Department of Chemistry, Indian Institute of Technology, Kharagpur, as a CSIR (Government of India) Fellow. At present, she is a Senior Research Fellow.

Biographical Information

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Debdutta Chakraborty received his B.Sc. (Chemistry Honors) and M.Sc. (Chemistry) degrees from the North Bengal University, West Bengal. Then he joined the research group of Professor P. K. Chattaraj for his Ph.D. degree in the Department of Chemistry, Indian Institute of Technology, Kharagpur, as a CSIR (Government of India) Fellow. At present, he is working as a Junior Research Fellow.