Computer simulation of the optical Kerr effect in liquid water

PhysicsLettersA 158 (1991) 216—226 North-Holland

PHYSICS LETTERS A

Computer simulation of the optical Kerr effect in liquid water M.W. Evans’ 433 Theory Center, Cornell University, Ithaca, NY 14853, USA Received 19 March 1991; accepted for publication 26 June 1991 Communicated by B. Fricke

A field applied molecular dynamics (FMD) computer simulation of the optical Kerr effect in liquid water has produced ferntosecond rise transients which reproduce accurately the available analytical theory, based on generalised Langevin—Kielich functions. The transients and associated field applied correlation functions prove to be sensitive to the anisotropy of polarisability in water, showing that femtosecond transient spectroscopy is potentially a much more accurate method of measurement than hitherto available.

1. Introduction We report the first computer simulation of the optical Kerr effect, using field applied molecular dynamics (FMD). FMD is a simple variation on standard molecular dynamics computer simulation in which the torque generated between the molecules of an ensemble and an external field is coded into the forces loop. FMD provides rise and fall transients, thermodynamic and structural data, and field-on or field-free statistical mechanical data, for example time correlation functions of many different kinds [1], some of which are Fourier transforms of directly observable spectra. FMD was devised originally for strong electric fields, and accurately reproduced known Langevin—Kielich functions [2—4]from the final levels of rise transients. Additionally, it gave details of the time resolution of the transients themselves on the femto/pico-second scale, information which is now becoming directly accessible experimentally [51 in the transient optical Kerr effect. FMD also gives a wealth of information from the same trajectories on the dynamics of the laser applied steady state in the optical Kerr effect, on fall transients [6—8],and on pair distribution functions [9]. The method was later extended for use with circularly polarised laser fields [10, 111, and for non-linear optical phenomena where in general [12—141,a laser field or field gradient forms a torque with an induced electric or magnetic dipole or multipole. FMD also revealed basic statistical mechanical laws such as fall transient acceleration [151, field decoupling [15], and rise transient oscillations on the femtosecond scale [16,171; its generality springing from the fact that the integral of the torque with respect to orientation is potential energy, a term which is added to the Hamiltonian as the starting point of analytical theory. The interaction of any type of field and any type of ensemble is therefore described at a fundamental level, as in semi-classical theory [18], but FMD provides additionally much more information from the same set of molecular trajectories. The particular type of torque relevant to the optical Kerr effect is described in section 2. The essentials of FMD in this context are recounted briefly in section 3, followed in section 4 by a description of rise transients and a comparison with generalised Langevin—Kielich functions. The latter from both simulation and theory are shown in two cases to change sign with anisotropy of polarisability, thus providing a potentially accurate method of measurement ofthis quantity in water from the transient optical Kerr effect [5]. Detailed agreement Present address: Institute of Physical Chemistry, University of Zurich, Winterthurerstrasse 190, CH 8057 Zurich, Switzerland.

216

0375-9601/9 1/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

Volume 158, number 5

PHYSICS LETFERS A

2 September 1991

is demonstrated between simulation and theory concerning this useful change of sign and other transient properties of the optical Kerr effect.

2. Torque of the opticai Kerr effect It is well known that the optical Kerr effect is observed [19,20] through the rotation of the plane of polarisation of a probe laser in response to a pulse from a pump laser. We consider a probe polarised linearly along the X axis of the laboratory frame (X, Y, Z). The pump pulse can be shown to set up the torque T1 =e2ve3x(a33 —a22)E~ ,

T~=e,xe3x(a,, —a33)E~ ,

T3 =e,~e2~(a22

—~

~)E~

(I)

in the frame (1,2,3) of the molecular principal moments of inertia of the water molecule. Here, axis 1 is the dipole, or C2,,, axis, axis 2 is perpendicular to the molecular plane, and axis 3 is mutually perpendicular in a right handed set. In eq. (1), E~is the square of the scalar electric field strength; amplitude (volts/metre) of the pump laser, and the quantities e,x, e2x and esx are X components of unit v~ctorsin axes 1,2 and 3. It is assumed that the polarisability is diagonalised in the same frame (1,2,3), giving the diagonal components a,,, a22, and a~ of the molecular polarisability of water. It is convenient in this context to define the two anisotropies of polarisability of the asymmetric top water molecule, (2) ö=~(a22—a33).

(3)

The torque components (1) are coded into the forces loop of any MD aIgorithm~and back transformed [12] into the laboratory frame (X, Y, Z) using a rotation matrix.

3. FMD method The FMD method was implemented with a sample of 108 water molecules interacting through a modified ST2 potential using Lennard-Jones and partial charge interactions, (H—H):

~/k=21.lK,

cr=2.25A,

(0—0):

~/k=58.4K,

a=2.80A,

qH=O.231e1

,

q(lonepair)=—0.231e1

,

q~=0.00~eI

(4)

This potential has been compared in the literature [211 with the ab initio MCYL, and with experimental data [22] over a very wide range of conditions. We stress that FMD can be used with any type of model or ab initio water potential, and any type of MD algorithm. With a time step of 0.5 fs, transients were evaluated for different E~,over a number of time steps sufficient for attainment of the final level, which was measured and used to construct generalised Langevin—Kielich functions (section 4) by FMD. The latter were also evaluated analytically [23—25].The transient generating stage was followed by FMD evaluation of statistical dynamical properties in the pump laser applied steady state, using running time averaging over a minimum of 6000 steps. A data band of many different examples was collected and used to characterise the field-on dynamics. Simulations were carried out at 296 K, 1.0 bar in the liquid state of water.

217

Volume 158, number 5

PHYSICS LETTERS A

2 September 1991

4. Rise transients and anisotropy of polarisability There have been numerous reports [26—32]of the anisotropy function y of water. Khanarian and Kent [261 have made a tabular comparison of experimental and ab initio estimates from various sources, six of whose entries were positive, and four negative. In this section we use three literature estimates of the diagonalised polarisability components of water, and compare for each case generalised Langevin—Kielich functions and time resolved rise transients from FMD and the available theory. These are given in table 1. We note that the experimental data in table 1 give a negative anisotropy y [26], and a conflicting positive anisotropy y [27]. We also use one ab initio result [28]. For each of these three data sets transients and generalised Langevin—Kielich functions (GLKs) were simulated for ,, and ,where n=2,4,6. GLKs were also evaluated analytically from [23—25] 2O) ~~exp[hcos2Ø (cos2O—l)]dØsin6d6 (5) ,f~cos~Oexp(qcos e,x f~exp(qcos2O)J~exp[hcos2Ø(cos26—l)]dØsinOdO —

—

— —

f~sin~Ocos~Øexp(qcos2O) J~exp[hcos2Ø (cos2O—l)] dØsinOdO f~exp(qcos2O)f~exp[hcos2Ø(cos2O—l)]dØsinOdO

(6)

.f~sin~Osin”Øexp(qcos2O) J~exp[hcos2Ø (cos2O—1)] f~exp(qcos2O)J~exp[hcos 20 (cos2O— 1)] dØsindØsin6dO GdO

(7)

where q=yE~/kT,

h_—~E~/kT.

(8)

These integrals were evaluated numerically using IBM software [33] based on fine-grid double Gauss—Legendre quadrature. Fig. 1 illustrates the fact that two of these GLK functions change sign with anisotropy of polarisability y, in the sense that they increase or decrease from an initial value. Points on these analytical GLKs show simulated data from FMD under the same conditions, with potential energy corresponding to the torque (1) computed over a minimum of 6000 time steps at each point. The FMD method successfully simulates the change of sign in and given analytically from (6) and (7) as y is changed from positive to negative. The function does not change sign analytically from eq. (5), and again this is reproduced successfully by FMD. These results show that (1) the sign of the anisotropy ofpolarisability of the water molecule canbe determined from an experimental measurement of the three GLKs described already, possibly with contemporary femtosecond transient optical Kerr effect apparatus [5]; (2) FMD is in detailed agreement with the analytical theory on GLKs.

Table I Literature estimates of the polarisability tensor ofwater diagonalised in (1,2,3). (In figs. 1—3, (a), (b) and (c) refer to the data sets of refs. [26], [27], and [28] respectively.) Ref. [26] [27] [28]

218

a 11

a22

a33

1.69 9.62 8.15

1.9 9.26 7.12

1.3 10.01 9.03

(S.1. units) (a.u.) (a.u.)

Volume 158, number 5

2 September 199 1

01

-lo a0

a2

_____ _ ________............... ___.__________ ..__ _ _..-... .._ ___. _&-

0.0

bl

a3

0.7

oa as

01

011

02

a1

M

m

so

a0

40

m

00

0

Fig. 1. Generaliscd Langevin-Kielich functions L,(q, h), Z&q, h), and L&q, /I) (labeled I, 2, 3, respectively) from analytical theory (curves) and field applied computer simulation (points) for liquid water at 293 K, I.0 bar: using the data (table 1) of (a) ref. [26], (b) ref. [27], and (c) ref. [28]. The functions initiate at l/3 (n=2), l/5 (n=4), and l/7 (n=6). Note that in two cases the GLKs change sign between data sets (a) and (b), respectively representing negative and positive y. (-_) n = 2, (- - -) n= 4, (- - . - ) n= 6. (9) FMD, n=2.

219

Volume 158, numberS

PHYSICS LETTERS A

uw.~o

2 September 1991

U

030 .-...... —~

0.9

035 023 027

0.1

0030.30

1’

am n..

-20

-tO

-18

-1.4

-1.2

-18

-0.8

-02 -0.4 -0.2

0.0

_._-~~.‘

-20 -16

-18

-14

-tO -10

-02 -02 -0.4

-0.2

0.0

a

0 u0~

140$

ci o20 0231 o~

0.7

02;t 0$ 0.51 1:

H

0.15 015

a. 0.15 0.15

0.3

O30j~ 037 025~/

coo 0.03 0.30

0.1

am

‘~

020

0.C

~~~—‘—-~——--—

0

I

2

0

3

4

5

0

Fig. I. Continued.

220

________________________ I 2 3 4 5 S 7 S 0

9

Volume 158, number 5

PHYSICS LETTERS A.

2 September 1991

UO~%4 0.33 0.24 031 030 0.20 035 027 035 025 0.24 023

am 021 020, 115 0.15 0.17

a. 0$ 024 0.12 0.12 0.11 020 0.24 0.00 0.07 0.24 026 0.04

am am

\

\

\

\,

\

\ ‘.,,~ “.,

am am0.00_____________ 0.06 020 023

‘-....

0.20—..--.-.-..0.25 0.30

0.35

0.40

0.46

Fig. 1. Continued.

5. Time resolution of optical Kerr effect rise transients The time resolution on the femtosecond level of the optical Kerr effect has been achieved [5] by KenneyWallace and co-workers. The data are in the form of rise transients of probe optical absorption as a pump pulse is passed through the ensemble. These data can be related to our GLKs through the birefringence of the optical Kerr effect, which is in turn related to the observable changes in power absorption coefficient [5] through the Kramers—Kronig equations.

It is possible therefore, using a numerical iterative scheme, for example, to determine the anisotropy of polarisability from the rise transient experimental data [5]. Furthermore, FMD gives the time resolution of the rise transient as illustrated in fig. 2 using the conflicting literature data sets of table 1. Fig. 2 shows that the time resolved transients are markedly different for each data set at fixed pump laser equivalent potential energy, obtained by integrating the torque over configuration space in the FMD algorithm. For the Khanarian— Kent estimate [26] there are no rise transient oscillations [16,17] known from Kramers theory [17] to be due to specific non-linearities in the molecular dynamics. The presence ofthese osçillations is observed clearly (fig. 2), however, using the data given by Zeiss and Meath [27] and the ab initio computation of van Hemert and Blom [28]. It is concluded that the details of the time dependence of the optical Kerr effect rise transient are intricate functions of the anisotropy of polarisability at a given pump laser intensity. It would be of interest to attempt to observe such oscillations experimentally [5]. Finally, we report that in the laser field applied steady state reached by the rise transient at a given pump laser intensity, a data bank of time correlation functions was accumulated to inve~tigatein detail the field-on statistical dynamics by FMD. The correlation functions were observed also to be dependent markedly on which data set of table 1 was used, and this is illustrated in fig. 3 for the orientational autocorrelation functions

221

Volume 158, number 5

PHYSICS LETTERS A

2 September 1991

ThN~~

0.1

...,.......,,

0.4

.

.

~

~

,~

,~r~

0.5

-‘

/

..-... ..,,../

.

ft

°,~

~

03

Fig. 2. Timeresolution oftransients for data sets of (a) ref. [261, (b) ref. [27 l~ and (c) ref. [28] (see table 1), simulated by FMD with equivalent laser potential energy of2.OkJ/mole. (—) ,(- - -) ,(_. — . —) ;i=l, 2,3 isthe label added to a, b, c in the figures.

222

Volume 158, number S

PHYSICS LETTERS A

2 September 1991

fl

_______________

0.5

:

0.5

02

b3

b2

oh

~

D~

~

Fig. 2. Continued.

223

Volume 158, number S

PHYSICS LETTERS A

2 September 1991

c~.A1I%

0.0

0.1

0.2

0.3

0.4

0.5

•

0.8

0.7 •

0.8

0.50 020 0.04 025 025 023 023

0.14 0.15 015

am

Fig. 2. Continued.

cc~A1~1L

~o r—~-..

O.S\

—

—‘-—_____

0.8 0.7

0.7

~1

A

°~ 0.4

H I\ i\ H H H H I\ flfl H ~ ~ I \

/\ 1\ / I \ / I / I \ II \I jJ I I \I \/ -O2HII\J\/ V II II ~! V H Ii V ~

0,1

~

03 01

03

03

~0,1

—0.1

-03

-0.4

-0.4

b

c

~0.5

-to 0.00

~

030 0.04 0.06 DOS 0.15 0.15 0.14 023 023 0.20

-to

________________________

020 025 0.04 020026020

023 0,14 0.15 030 023

Fig. 3. As for fig. (2), orientational autocorrelation functions in the post transient steady state for data sets of (a) ref. [26], (b) ref. [27], and (c) ref. [28] (see table 1): (—) i=j~’X;(- - -) i=j= Y; (— —. —) i=J=Z.

224

Volume 158, numberS

C11~(t)=”2”2’

PHYSICS LETTERS A

2 September 1991

i=i.

(9)

Such differences were also observed for the time correlation functions of molecular angular momentum

c

— ____________

(10)

and of rotational velocity

c

— ______________

(11)

Off diagonal components (cross correlation functions) were found to change sign with the anisotropy of p0larisability y. Further detailed discussion of these extensive data will be reported elsewhere. The Fourier transform of the rotational velocity autocorrelation function [34] is the far infrared power absorption coefficient, and were it possible to observe this spectrum in the presenceof a pump laser pulse, FMD as used in this paper shows that it would give information on the molecular dynamics of the optical Kerr effect in the laser on steady state, and specifically on the anisotropy of polarisability.

Acknowledgement MWE thanks Professor Dr. Georges Wagniere for an invitation to the University of Zurich as a Guest of the University, and for many interesting discussions. The Swiss NSF is thanked for the award of a Senior Visiting Fellowship. ETH Zurich is acknowledged for a major grant of computer time on the IBM 3090 supercomputer on which this work was carried out. Dr. Laura J. Evans is thanked for invaluable help with the SAS plotting facility of the Irchel mainframe, used to build up the data banks. Last but not least, Professor S. Wozniak is thanked for many interesting discussions and advice on torques and generalised Langevin—Kielich functions.

References [I] [2] [3) [4) [5) [6)

M.W. Evans, in: Advances in chemical physics, Vol. 81, eds. I. Prigogine and S.A. Rice (Wiley—Interscience, New York, 1991). S. Kielich, in: Dielectric and related molecular processes, Vol. 1, Sr. rep. M. Davies (Chem. Soc., London, 1972). M.W. Evans, J. Chem. Phys. 76 (1982) 5473, 5480. M.W. Evans, J. Chem. Phys. 77 (1982) 4632; 78 (1983) 925, 5403. C. Kalpouzos, D. McMorrow, W.T. Lotshaw and G.A. Kenney-Wallace, Chem. Phys. Lett. 150 (1988) 138. M.W. Evans, P. Grigolini, 0. Pastori, I. Prigogine and S.A. Rice, eds., Advances in chemical physics, Vol. 62 (Wiley—Interscience, New York, 1985). [7]J. Jortner, R.D. Levine, I. Prigogine and S.A. Rice, eds., Advances in chemical physics, Vol. 47 (Wiley—Interscience, New York, 1981). [8]M.W. Evans, Phys. Scr. 30(1984)91. [9] E. Clementi, ed., MOTECC 89 and 90 (Escom, Reidel). [10] M.W. Evans, G.C. Lie and E. Clementi, Chem. Phys. Lett. 138 (1987) 149. [11] M.W. Evans, G.C. Lie and E. Clementi, J. Chem. Phys. 87 (1987) 6040. [1210. Wagniere, Phys. Rev. A 40 (1989) 2437. [13] D.C. Hanna, M.A. Yuratich and D. Cotter, Non-linear optics of free atoms and molecules (Springer, Berlin, 1979). [14] M.W. Evans and G. Wagnière, Phys. Rev. A 42 (1990) 6732. [IS] P. Grigolini, in: Advances in chemical physics, Vol. 62, eds. M.W. Evans, P. Grigolini, G. Pastori, I. Prigogine and S.A. Rice (Wiley—Interscience, New York, 1985).

225

Volume 158, number 5

PHYSICS LETTERS A

2 September 1991

[16] M.W. Evans, G.C. Lie and E. Clementi, Phys. Lett. A 130 (1988) 289. [17) W.T. Coffey, in: Advances in chemical physics, Vol. 83, eds. M.W. Evans, I. Prigogine and S.A. Rice (Wiley—Interscience, New York, 1985). [18] L.D. Barron, Molecular light scattering and optical activity (Cambridge Univ. Press, Cambridge, 1982). 19] S. Wozniak, Mol. Phys. 59 ( 1986 ) 42 1. [20] H.J. Coles and B.R. Jennings, Mol. Phys. 31(1976) 570. [21] M.W. Evans, K.N. Swamy, K. Refson, G.C. Lie and E. Clementi, Phys. Rev. A 36 (1988) 3935. [22] M.W. Evans, G.C. Lie and E. Clementi, J. Chem. Phys. 88 (1988) 5157. [231 S. Wozniak and R. Zawodny, Phys. Lett. A 85 (1981) 111. [24] S. Wozniak, B. Linder and R. Zawodny, J. Phys. (Paris) 44 (1983) 403. [25]S.Wozniak,Phys.Lett.A 148 (1990) 188. [26] G. Khanarian and L. Kent, J. Chem. Soc. Faraday Trans. II 77 (1981) 495. [27] G.D. Zeiss and W.J. Meath, Mol. Phys. 30 (1975) 161. [28] M.C. van Hemert and C.E. Blom, Mol. Phys. 43 (1981) 229. [29] M.S. Beevers and G. Khanarian, AustralianJ. Chem. 33 (1980) 2885. [30] W.F. Murphy, J. Chem. Phys. 67 (1977) 5877. [31] R.K. Kilanno, E. Dempsey and 0. Parry Jones, Chem. Phys. Lett. 53 (1978) 542. [32] E.A. Moelwyn Hughes, Physical chemistry (Pergamon, Oxford, 1964) pp. 384, 473. [33] IBM ESSL Library, routine DGLNQ2.

226