Multiphoton vibration-rotation excitation of the OH molecule in the presence of an infrared laser beam

Volume

128, number

1

CHEMICAL

PHYSICS

11 July 1986

LETTERS

MULTIPHOTON VIBRATION-ROTATION EXCITATION IN THE PRESENCE OF AN INFRARED LASER BEAM

OF THE OH MOLECULE

Man MOHAN Department of Applied Maihematics and TheoretIcal Physics, The Queen’s Universrty of Belfast, Belfast BT7 INN, UK

and Bhupat SHARMA Department of Physics and Astrophysrcs, University of Delhr, Delhi 110007, India Received

15 January

1986; in final form 25 April 1986

The Floquet method (non-perturbative) is used to investigate the multiphoton vibration-rotation excitation of the OH molecule in the presence of an infrared laser beam. A number of interesting features, such as line broadening and the dynamic Stark shift, are studied as a function of laser intensity and frequency.

1. Introduction

Recently there has been considerable attention to the study of multiphoton excitation of atoms and molecules [l] due to its importance in the field of laser-induced chemistry [2,3] and the production of powerful lasers. Also there has been growing interest in the study of the atmosphere and stellar and interstellar spaces with laser probes due to their efficient and effective methods. In this paper we study the OH molecule as it is an important radical which is formed in many elementary chemical processes in the atmosphere [4] , in combustion processes [5] , in stellar [6] and interstellar [7] space. Here we use the Floquet theory (non-perturbative) for the calculation of multiphoton vibration-rotation spectra of OH [8,9]. The Floquet theory [8,9] relates the solution of the S&&linger equation with a periodic Hamiltonian (laser + molecular system) to the solution of another Schrijdinger equation with a time-independent Hamiltonian represented by an infinite matrix (called a Floquet matrix). We use this theory to calculate the effect of intensity and frequency of the laser field on the multiphoton vibration-rotation spectra of the OH molecule. In section 2 we present a brief survey of the Floquet theory. In section 3 we discuss the various parameters and computation details, and in section 4 we consider the results.

Here we consider the interaction of a diatomic molecule with a laser beam defined by E = Eo cos(ot), where IEoIis the amplitude of the field and w is the frequency of the laser beam. The Schriidinger equations for such a system can be written (in au) as

a*,tr,0 = A\kp(r,t) ,

i - at

where fi = fro - l(r) E, cos(ot) cos 8 ; I?,-, is the Hamiltonian for the isolated diatomic molecule, p(r) is the mo38

0 009-2614/86/s 03.50 0 Elsevier Science publishers B.V. (Norht-Holland Physics Publishing Division)

CHEMICALPHYSICSLETTERS

Volume128, number 1

11 July 1986

lecular dipole moment function, r is the internuclear distance and 8 is the angle between the polarization vector of the laser beam and the body-fixed molecular axis. We can expand the total wavefunction q,(r,t) of the system in terms of the field-free molecular eigenfunctions &(r) which satisfy the equation

with the initial condition Qp(r, t = 0) = @r) .

(4)

As x (r) represents the molecular eigenfunction, we can represent it as a product of vibrational Z”(r) and rotational fi* J(e, 4) wavefunctions. So, in brief, the subscript 4 represents vibrational u and rotationalj, mi quantum number!. When the tield is turned on, u andi do not remain good quantum numbers, but the angular momentum projection quantum number m,-does remain a good quantum number because the interaction p E in eq. (1) is independent of 9, the azimuthal angle. Substituting eq. (3) into eq. (1) and using the orthogonahty conditions for the molecular states, we obtain l

where F@ = (x,(r) I/de cm 0 I x&P

(6)

is a dipole matrix element. In matrix notation the above equation can be written as

(7)

iF=H,(r)*@),

where H,(t) is a periodic function of time with period T = 27r/w. According to Floquet theory [8,9], eq. (7) admits the solution AF(t) =4,(t) exp(-ipt)

,

(8)

where Mf) is a periodic function of time over a complete cycle T and satisfies $(t + 7) = 9(t)

(9)

and c is a diagonal matrix called the characteristic exponent. According to Shirley [9], the Fourier expansions of AF(t) and H,(t) are given by AF(t) =

(10)

(11) where frJand 3C,“sare the Fourier amplitudes corresponding to particular value of m. Here the indices r and I range over the molecular states with dimension N=N,,Nr (N, is the number of vibrational states, N, the number of rotational states) and the Fourier index m indicates the photon number. 39

11 July 1986

CHEMICAL PHYSICS LETTERS

Volume 128, number 1

Table 1 Molecular parameters for the OH molecule in the electronic ground state R(A)

&cm-’ )

cr,(cm-’ 1

w,(cm-‘1

wexe(cm-’ 1

D(ev)

0.971

18.87

0.714

3739.3

86.4

4.40

Eqs. (10) and (11) are substituted into eq. (7) to yield the eigenvalue equation for the Fourier components Cl and the characteristic values ccl.The eigenvalue problem is

Once the eigenvectors frlf corresponding to the eigenvalue /.11are found by diagonalizing the Floquet matrix HP, the transition amplitude a,(t) can be determined from eq. (10). More explicitly, the transition probability from the initial vibrational state p = 1t+mj>(with mi staying constant in time) to the final state 4 = 1u!mi>is given by [8,9] Pp+&, to) =PZL “‘j’(t, to) = I(u’i’IU(t,t&I?12 = la”f+vy(t,f())V ,

(13)

where U(t, to) is the evolutron operator. Of most experimental interest, averaging eq. (13) over the initial time to, while keeping the elapsed time (t to) fixed, yields

pi& - to)=G 3

If;)

exp[-ipl(t -

to)]

f$P .

(14)

For continuous coherent operation of the laser we average the above equations over t - to = T, and obtain the long-time average transition probability as &#,

~)‘=~;&,(I,

o) = tin_ T-l ST Pp-+&) dr = q 0

5

Ifp”Jf;,j2 .

(15)

In section 3 we describe the procedure to form the Floquet matrix and the determination of the long-time average transition probability.

3. Parameters and computations Here we have taken the example of the OH molecule. In constructing the Floquet Hamiltonian HP (the term within the brackets of eq. (12)), we require the evaluation of the dipole matrix element (eq. (5)) between the adjacent levels [8] , that is, V48 =v =

VjtTlp.Q’Ptlj

~(~(uI(r

-

~~)lu’)~,

=(dmjl&) {[o

+

COSBlU~‘??lj)=&6,*

1)2 - m;] /(2j + 1) (2j + 3)}lj2,

=pl(uI(r-re)lu’)re [Cj2-mf)/@- 1)(2j+1)11i2 ,

j’ = j + 1 ;

i’=i-1.

(16)

The angular momentum projection quantum number mj remains a good quantum number when the field is turned on. We have used a recent calculation [lo] for the evaluation of accurate vibrational dipole matrix elements for the OH molecule. The anharmonic wavefunctions, needed to calculate vibrational dipole matrix ele40

Volume 128, number 1

CHEMICAL PHYSICS LETTERS

11 July 1986

ments, have been obtained by solving numerically the radial Schrodinger equation as described by Cooley [ 1I] . In order to minimize the errors in matrix elements due to inaccuracies of the potential energy function, use has been made of the RKR potential for the OH molecule [ 121. The spectroscopic parameters [ 13] are defined in table 1. Due to the occurrence of the OH molecule in nature in lower vibrational states [14] and due to the availability of reliable molecular parameters together with accurate dipole matrix elements, we have considered 15 molecular vibrator-rotator states distributed corresponding to the rotational states in each vibrational manifold (.5,5,5). In our calculations we have truncated the Floquet Hamiltonian Zf to contain seven Floquet photon blocks (n = 0, +l, +2, +3) although the results were found to, be converged after truncating the Floquet Hamiltonian by including only five photon blocks (n = 0, +l , *2). We can thus represent the Floquet Hamiltonian, defined by eq. (12), by a block tridiagonal Hermitean matrix with rows and columns denoted by the trio of indices u, j and n r \

n=3

“A+ 301 B

n=2

n=l

n=o

n=-1

n=-2

n=-3

B

n=3

At201

B

B

Atol

B

B

A

B

B

A-wl

B

B

A-201

B

B

A - 3o~L.,,

H, =

n=2 n=l n=O

3

(17)

n=-1 w-2

J

n=-3

where I is a unit matrix, A = Eui 6,t ~5~~is a diagonal matrix and Eui is the rovibrational energy of the state. Further, in the above equation B = &j,v~i~= i lEo I Voj,viI l is an off-diagonal matrix; IEo I is the laser field amplitude and Vvi u 7’ is a dipole matrix defined in eq. (16). Using the standard diagonalization routines the Floquet Hamiltonian HF is diagonal&d to yield the characteristic Floquet exponents ~1 and corresponding Fourier components fm defined by eq. (12). These are then substi‘rf. . tuted into eq. (15) to yield the infinite-time average transition proba rhty pV&,j, for transition from the initial vibrator state IuQ> to the final vibrator state Iu’Z’mj>.

4. Results and discussion Under most experimental conditions (i.e. at or below room temperature) the molecules are populated only among the rotational states of the ground vibrational state. Also studies of the dynamics of transitions evolved from various u = 0 andi states are essential to many spectroscopic investigations. Therefore in this section we discuss long-time average transition probabilities for the OH molecule initially prepared in a specific Iu = 0,j = 0, mj = 0) state. In fig. 1 we have shown the variation of the long-time average transition probability Z$o+l 1 for transition fromlu=O,~=O;~~=O~to~u=l,~=l;~~=O~withlaserfrequen~w(incm-1)andlaser~~ensityZ(inW cmb2). It is clearly seen from fig. 1 that the probability at Z = lOlo W cme2, increases quite steadily from w = 3598 to w = 3601 cm-l where it has an extreme value. However with further increase in the frequency, the probability ‘8 ojl 1 decreases. At a,lower intensity Z = lo8 W cmm2 (also shown in fig. l), the transition probability rises quite’sharply around the frequency o = 3601 cm-l and has an extreme value near w = 3602.09 cm-l. 41

Volume 128, number 1

11 July 1986

CHEMICAL PHYSICS LE’ITERS

203.. I%! 6 0’2-,

0%

----3598

3599

Fig. 1. T~e~~er~~

3600

3601

3602 3603 I?4 -__^_.^..-*,__-t Kt5JUkNLT i cm I

transition probability F&,+,

,

3605

3606

3607

3608

t versus laser frequency w (in cm-“) for the OH molecuie.

Thus the shifting of resonance by u = 0.11 cm -l with increase of laser power by lo2 W cmw2 is a clear indication of the dynamic Stark shift which increases with increasing intensity of the laser beam. Further, the fwhm increases from 0.1 to 1.1 cm-l as the intensity increases from I = lo8 to I= lOlo W cm-2, which indicates that the line broadening is very prono~~d for the OH molecule and increases with increase in the intensity.

FREQUENCY

(cd)

Fig. 2. Same as fg_ 1 but for transition Iv = O,] = 1, mr = 0) to lu = l,i = 2; rnj = O), that is,F&

42

*.

t

Volume 128, number 1

11 July 1986

CHEMICAL PHYSICS LE’ITERS

0’2 --

3664

3665

3666

3667

3668

3669

FREQUENCY Fig. 3. Same as fii. 1 but for transition

3695

3686

is697

ia

3670

3671

3672

Iv = 0, j = 2; mj = 0) to Iv = 1, j = 2; mi = O),that is,?&

3699

3700

FREQUENCY

3673

3674

km-‘)

3701

h

3702

3703

3.

7

3704

3705

(cm?)

Fig. 4. Same as fii. 1 but for transition Iv = 0, j = 3; mi = 0) to Iu = 1, j = 3; mj = O), that is, F$+,

4

We have plotted the other transitions in figs. 2-4, that is,P7,1_I 2, Pg 2_,1 3 andPi I+1 4 versus frequency for I = 108 and I = lOlo W cmb2. AlI these plots show the usual Lor&tzi& &a&s like-fi. 1. \ire have thus found that the dynamic Stark shift and line broadening play an important role during vibrotational transitions of the OH molecule in the presence of an intense laser beam. 43

Volme

128, number 1

CHEMICAL PHYSICS LETTERS

11 July 1986

Acknowledgement The authors are grateful to Professor S.N. Biswas for constant encouragement during the course of this work. This work is supported by DST, India.

fl] N. Bloemberge~ and E. Yablonovitcb, Phys. Today 31 (1978) 23; P.A. Schulz, Aa.S. Sudbo, D.J. Kxajnovich, H.S. Kwok, Y.R. Shen and Y.T. Lee, Ann. Rev. Phys. Chem. 30 (1979) 379; V.S. Letokhov and A.A. Makarov, Soviet Phys. Usp. 241 (1981) 366; P. Lambropoulos, Advan. At. Mol. Phys. 12 (1976) 87; J. Morellec, D. Normand and G. Petite, Advan. At. Mol. Phys. 18 (1982) 97. [2] T.F. George, J. Phys. Chem. 86 (1982) IO; F.H. Mies and D. Henderson, eds., Theoretical chemistry, advances and perspectives, Vol. 8 (Academic Press, New York, 1980); P. Hering, P.R. Brooks, R.F. Curl Jr., R.S. Judson and R.S. Lowe, Phys. Rev. Letters 44 (1980) 648; T.C. Maguire, P.R. Brooks and R.F. Curl Jr., Phys. Rev. Letters 50 (1983) 1918. [ 31 M. Mohan, J. Chem. Phys. 25 (1981) 1250; M. Mohan, K.F. Milfeld and R.E. Wyatt, Chem. Phys. Letters 99 (1983) 411; M. Mohan and P. Chand, J. Chem. Phys. 71(1979) 2207; M. Mohan, P. Chand and B. Sharma, Mol. Phys. 54 (1985) 959; B. Sharma and M. Mohan, Chem. Phys. Letters 118 (1985) 553; J.C. Peploski and L. Eno, J. Chem. Phys. 83 (1985) 2947. [4] A.V. Jones, Space Sci. Rev. 1.5 (1973) 355; D.J.W. Kendall and T.A. Clark, J. Quant. Spectry. Radiat. Transfer 21 (1979) 511. [5] J.A. Coxan and SC. Foster, Can. J. Phys. 60 (1982) 41. [6] W.J. Wilson and A.H. Barrett, Science 161 (1968) 778. [7] BJ. Robinson and R.X. McGee, Ann. Rev. Astron. Astrophys. 5 (1967) 183. [8] SC. Leasure and R.E. Wyatt, Chem. Phys. Letters 61 (1981) 6197; J.V. Mobney and F.H.M. Faisal, J. Phys. B12 (1979) 2829; J.V. Moloney and W.J. Meath, Phys. Rev. Al7 (1978) 1.550. [9] J.H. Shirley, Phys. Rev. B138 (1965) 979. [ 10 ] H.J. Werner, P. Rosmus and E.A. Reinsch, J. Chem. Phys. 79 (1984) 905. [ 111 J .W. Cooley, Math. Comp. 15 (1961) 363. [12] R.J. FaBon, I. Tobii and J.T. Vanderslice, J. Chem. Phys. 34 (1961) 167. [13] K.P. Huber and G. Herzberg, Constants of diitomic molecules (Van Nostrand Reinhold, New York, 1979). [ 141 D.C. Clary, Mol. Phys. 46 (1982) 1099.

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