Chattaraj P.K. (Ed.) Quantum Trajectories

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224 Quantum Trajectories

A time dependent (TD) variant of DFT is also available [2] which allows one to invert

the mapping between the TD external potential and the TD density up to an additive

TD function. All TD properties are unique functionals of density ρ(r, t) and current

density j(r, t ). The TDDFT may be used through the solution of a set of TD Kohn–

Sham type equations [2,3]. Alternatively, one may resort to a many-particle version

of Madelung’s quantum ﬂuid dynamics (QFD) [4] to obtain the basic QFD variables

ρ(r, t ) and j(r, t ) and in turn all other TD properties. The backbone of QFD comprises

two equations, viz., an equation of continuity,

∂ρ(r, t)

∂t

+∇.j(r, t) = 0 (14.4a)

and an Euler type equation of motion,

∂j(r, t)

∂t

= P

υ(r,t )

[ρ(r, t ), j(r, t)]. (14.4b)

Unfortunately TDDFT does not provide the exact form of the functional

v(r,t)

[ρ(r, t ), j(r, t)]. A quantum ﬂuid density functional theory (QFDFT) [5] has

been developed to tackle this problem by introducing an approximate version of this

functional, writing the many-particle QFD equations as

∂ρ

∂t

+∇.

(

ρ∇ξ

)

= 0 (14.5a)

and

∂ξ

∂t

(∇ξ)

δG[ρ]

δρ



ρ(r



, t)

|r −r



+ υ

ext

(r, t ) = 0. (14.5b)

In the above equations ξ is the velocity potential and the universal Hohenberg–Kohn

functional G[ρ] is composed of kinetic and exchange-correlation energy functionals,

while υ

ext

(r, t ) contains any TD external potential apart from υ(r). These two equa-

tions and a 3D complex-valued hydrodynamical function Φ(r, t) provide the QFDFT

whose basic equation is the following generalized nonlinear Schrödinger equation

(GNLSE):



−

∇

+ υ

eff

(r, t )



Φ(r, t ) = i

∂Φ(r, t)

∂t

, i =

√

−1 (14.6a)

where

Φ(r, t ) = ρ(r, t)

exp[iξ(r, t)] (14.6b)

ρ(r, t ) = Φ

∗

(r, t )Φ(r, t) (14.6c)

j(r, t ) =[Φ

∇Φ

− Φ

∇Φ

]=ρ∇ξ (14.6d)

and the effective potential υ

eff

(r, t ) is given by

eff

(r, t ) =

δT

[ρ]

δρ

δE

[ρ]

δρ



δρ(r, t )

|r −r



+ υ

ext

(r, t ), (14.6e)

[ρ] and E

[ρ] being respectively the non-Weizsäcker contribution to the kinetic

energy and the exchange-correlation energy functionals.

Chemical Reactivity Dynamics 225

The phase of the hydrodynamical function Φ(r, t)[ξ(r, t )], Equation 14.6b, may

be used to deﬁne a velocity (Equation 14.6d) as

r =∇ξ(r, t)|

r=r(t)

(14.7)

which will govern the motion of a point-particle guided by a pilot wave described by

Φ(r, t ), like the wavefunction Ψ(r, t), within the quantum theory of motion (QTM) [6].

Now an ensemble of particle motions may be generated by considering different ini-

tial positions of the particles guided by the same wave and satisfying the constraint

that ρ(r, t)dr gives the probability that a particle belongs to an ensemble distributed

between r and r +dr at time t, where ρ(r, t) is given by |Φ(r, t)|

(Equation 14.6c).

Thus the solution of Equation 14.7 or equivalently the associated Newtonian equa-

tion of motion with different initial positions would yield the so-called quantum or

Bohmian trajectories [7].

The particle motion in QTM and the ﬂuid motion in QFD take place under the

inﬂuence of the external classical potential augmented by a quantum potential V

(r, t )

written as

(r, t ) =−

∇

(r, t )

. (14.8)

The quantum domain behavior of classically chaotic systems like anharmonic oscil-

lators, ﬁeld induced barrier penetration in a double well potential, chaotic ionization

in Rydberg atoms, etc. has been analyzed [8] using QFD and QTM mainly in terms

of the distance between two initially close quantum (Bohmian) trajectories and the

related quantum Lyapunov exponent and Kolmogorov–Sinai entropy.

14.2 CHEMICAL REACTIVITY DYNAMICS

Chemical reactivity can be analyzed through various global and local reactivity

descriptors deﬁned within a conceptual DFT framework [9]. The chemical poten-

tial (Equation 14.2) may be expressed as [10],

µ =−χ =



∂E

∂N



v(r)

(14.9)

where χ is the electronegativity. Correspondingly, the second derivative deﬁnes the

hardness [11], viz.,

η =



∂

∂N



v(r)



∂µ

∂N



v(r)

(14.10)

or equivalently [12]

η =



η(r, r



)f (r



)ρ(r)drdr



(14.11)

where η(r, r



) is the hardness kernel [12] and f (r) is the Fukui function [13]. Elec-

trophilicity is deﬁned as [14]

ω =

2η

. (14.12)

226 Quantum Trajectories

Several electronic structure principles have been introduced to understand these

descriptors better. During the molecule formation process electronegativity gets

equalized. According to the electronegativity equalization principle [15] the molec-

ular electronegativity is roughly the geometric mean of the corresponding isolated

atom values. During an acid–base reaction, hard likes hard and soft likes soft as

per the statement of the hard–soft acid–base principle [16]. Moreover, the maximum

hardness principle (MHP) dictates that [17], “there seems to be a rule of nature that

molecules arrange themselves so as to be as hard as possible.” Owing to the inverse

relationship [18] between hardness and polarizability a minimum polarizability prin-

ciple has also been proposed [19] as, “the natural direction of evolution of any system

is toward a state of minimum polarizability.” A minimum electrophilicity principle is

also known [20].

In order to understand the dynamical implications of these electronic structure

principles the GNLSE has been solved for two TD processes and the time evolution

of various reactivity parameters has been analyzed [21]. The chosen TD processes

include: (a) ion–atom collision and (b) atom–ﬁeld interaction, known as two of the

most popular models to simulate a chemical reaction.

A TD chemical potential may be deﬁned as [22]

µ(t) =

δE(t)

δρ

|∇ξ|

δT

δρ



ρ(r



, t)

|r −r



δE

δρ

+ v

ext



, t) (14.13)

which is equal to the total electrostatic potential at a point r

(a measure of the covalent

radii of atoms in the ground state [23]) where the functional derivative of the total

kinetic energy balances that of the exchange-correlation energy, viz.,



|∇ξ|

δT

δρ

δE

δρ



r=r

= 0. (14.14)

Therefore,

µ(t) =−χ(t ) =



ρ(r, t )

− r|

dr +v

ext

, t). (14.15)

In the ion–atom collision problem, the dynamic µ proﬁle (Figure 14.1) divides the

whole collision process into three distinct regimes, viz., approach, encounter, and

departure [21a]. The qualitative features of these proﬁles for various electronic states

(

F) of different helium isoelectronic systems (He, Li

,Be

only two states for C

) remain the same in the scale we plot. In the encounter

regime, where the actual chemical reaction takes place, hardness gets maximized

[21a] (Figure 14.2) and polarizability gets minimized [21a] (Figure 14.3). Moreover,

for a hard–hard interaction (say between He and H

) the maximum hardness value is

the largest and the minimum polarizability value is the smallest when compared with

the other similar interactions [21a]. These results justify the validity of the associated

electronic principles in a dynamical context.

During the interaction between an atom and an external axial electric ﬁeld there

starts a competition between the atomic nucleus and the external ﬁeld to govern the

electron-density distribution. Figure 14.4 depicts the external ﬁelds of different colors

Chemical Reactivity Dynamics 227

–7.0

–3.5

0.0

010200

–7.0

–3.5

0.0

–7.0

–3.5

0.0

0102001020

–7.0

–3.5

0.0

–7.0

–3.5

0.0

10 20

01020

µµ

FIGURE 14.1 Timeevolution of chemical potential (µ, au) during a collision process between

an X-atom/ion (X = He, Li

,Be

) in its ground state and a proton. (Reprinted

with permission from Chattaraj, P. K. and Maiti, B., J. Am. Chem. Soc., 125, 2705, 2003.

9.8 10.0 10.2

0.00E+000

2.00E+008

4.00E+008

9.8 10.0 10.2

0.00E+000

2.00E+008

4.00E+008

9.8 10.0 10.2

0.00E+000

2.00E+008

4.00E+008

9.8 10.0 10.2

0.00E+000

2.00E+008

4.00E+008

9.8 10.0 10.2

0.00E+000

2.00E+008

4.00E+008

ηηη

ηη

FIGURE 14.2 Time evolution of hardness (η, au) during a collision process between an

X-atom/ion (X = He, Li

,Be

) in various electronic states (

F) and

a proton. (······)

S; (−−−)

P; (-·-·-·-)

D; (——)

F. (Reprinted with permission from

Chemical Society.)

228 Quantum Trajectories

81012

81081012

81012

αα

FIGURE 14.3 Time evolution of polarizability (α, au) during a collision process between an

X-atom/ion and a proton. See the caption of Figure 14.2 for details. (Reprinted with permission

American Chemical Society.)

–0.0000010

–0.0000008

–0.0000006

–0.0000004

–0.0000002

0.0000000

0.0000002

0.0000004

0.0000006

0.0000008

0.0000010

–0.010

–0.008

–0.006

–0.004

–0.002

0.000

0.002

0.004

0.006

0.008

0.010

–0.0000010

–0.0000005

0.0000000

0.0000005

0.0000010

–0.010

–0.005

0.000

0.005

0.010

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

–100

–80

–60

–40

–20

100

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

–100

–80

–60

–40

–20

100

ttt

= 1.0e–06

ε2ε1

= 0.01 ε

= 100

FIGURE 14.4 Time evolution of the external ﬁeld: ε1 (—-) monochromatic pulse; ε2(—

–) bichromatic pulse. Maximum amplitudes: ε

= 10

−6

, 0.01, 100; ω

= π, ω

= 2ω

(Reprinted with permission from Chattaraj, P. K. and Maiti, B., J. Phys. Chem. A, 105, 169,

Chemical Reactivity Dynamics 229

–12.727030

–12.727028

–12.727026

–12.727024

–12.727022

–12.727020

–12.727018

–12.727016

–12.727014

–12.727012

–12.727010

–12.727008

–12.727006

–12.727004

–12.727002

–12.727000

–13.0

–12.9

–12.8

–12.7

–12.6

–12.5

2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20

–150

–100

–50

100

150

–0.9563015

–0.9563010

–0.9563005

–0.9563000

–0.9562995

–0.9562990

–0.9562985

–1.00

–0.98

–0.96

–0.94

–0.92

–0.90

2468101214161820 2468101214161820 682 4 10 12 14 16 18 20

–150

–100

–50

100

150

ttt

= 1e–06

µµ

= 0.01 ε

= 100

FIGURE 14.5 Time evolution of Chemical potential (µ, au) when a helium atom is subjected

to external electric ﬁelds (GS, ground state; ES, excited state): (—-) monochromatic pulse;

(···) bichromatic pulse. Maximum amplitudes: ε

= 10

−6

, 0.01, 100; ω

= π, ω

= 2ω

(Reprinted with permission from Chattaraj, P. K. and Maiti, B., J. Phys. Chem. A, 105, 169,

and intensities. After the initial transients die out the TD chemical potential oscillates

in phase (Figure 14.5) with the external ﬁeld. However, hardness being a second

order response requires a stronger ﬁeld to exhibit in-phase oscillations (Figure 14.6).

In general the ground state hardness is larger than the corresponding excited state value

while electrophilicity shows the reverse trend [21c,d], as expected. These descriptors

may be used in analyzing the chaotic ionization in Rydberg atoms [24]. The in-phase

oscillations in chemical potential continues (Figure 14.7) even in the presence of a

super-intense ﬁeld (intensity = 3.509 × 10

W/cm

) which is, however, getting

disturbed in the case of hardness (Figure 14.8).

For a weak external ﬁeld the central Coulomb ﬁeld due to the nucleus will provide

a pulsating atom with roughly a spherical electron-density distribution. However, as

the ﬁeld intensity increases there will be a cylindrical density distribution and the

atom will behave as an oscillating dipole which will emit radiation including higher

order harmonics [21d].

In summary, conceptual DFT provides quantitative deﬁnitions for different reac-

tivity descriptors whose time evolution during a TD process like an ion–atom colli-

sion or an atom–ﬁeld interaction may be analyzed within a QFDFT framework. The

230 Quantum Trajectories

FIGURE 14.6 Time evolution of hardness (η) when a helium atom is subjected to external electric ﬁelds (GS, ground state; ES, excited state): (-)

monochromatic pulse; (···) bichromatic pulse. Maximum amplitudes: ε

= 10

−6

, 0.01, 100; ω

= π, ω

= 2ω

. (Reprinted with permission from

Chemical Reactivity Dynamics 231

246810

–20,000

–15,000

–10,000

–5000

5000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

50,000

246810

–100

–80

–60

–40

–20

100

Intensity_3.509 × 10

; ω

= π

Intensity_3.509 × 10

; ω

= 2π

Intensity_3.509 x 10

FIGURE 14.7 Time evolution of chemical potential (µ, au) when a helium atom in its ground

state is subjected to external electric ﬁelds: Intensities: I = 3.509 × 10

W/cm

, 3.509 ×

W/cm

, 3.509 × 10

W/cm

, 3.509 × 10

W/cm

; ω

= π, ω

= 2π.

0246810

Intensity_3.509 × 10

246810

7.27406

7.27408

7.27410

7.27412

7.27414

Intensity_3.509 × 10

, ω

= π

Intensity_3.509 × 10

, ω

= 2π

Intensity_3.509 × 10

FIGURE 14.8 Time evolution of hardness (η, au) when a helium atom in its ground state is

subjected to external electric ﬁelds. Intensities: I = 3.509×10

W/cm

, 3.509×10

W/cm

3.509 × 10

W/cm

, 3.509 × 10

W/cm

; ω

= π, ω

= 2π.

232 Quantum Trajectories

necessary TD density and current density are obtained through the solution of the per-

tinent generalized nonlinear Schrödinger equation. Dynamical variants of different

electronic structure principles manifest themselves.

ACKNOWLEDGMENTS

We thank CSIR, New Delhi for ﬁnancial assistance and Professors R. G. Parr, B. M.

Deb, S. K. Ghosh and A. S. Sanz and Drs. S. Nath, S. Sengupta, A. Poddar, B. Maiti,

and D. R. Roy for their help in various ways.

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