J. Phys. Chem. 1083, 87, 735-739
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ARTICLES Comparison of the Fioquet and Rotating-Wave Methods for Multiphoton Excitation of
SF, D. C. Clary Department of Chemistry, Universlty of Manchester Institute of Science and Technology, Manchester M60 lQD, UK (Received: August 23, 1982; In Final Form: October 12, 1982)
The Floquet method for calculating exact molecular multiphoton transition probabilities for a given Hamiltonian is compared to the rotating-wave approximation (RWA). The two methods are used in calculations on the multiphoton excitation of the v3 vibrational mode of SF,+ An accurate, three-dimensional molecular Hamiltonian is employed. The rotating-wave and Floquet results are in excellent agreement.
1. Introduction The multiphoton excitation of molecular vibrational modes using intense infrared laser radiation has been the subject of much research in recent years. The SF6molecular, in particular, has been studied in many experiments.l These experiments, however, have not yet reached the stage of sophistication in which they can provide accurate information on the detailed mechanism of multiphoton excitation. Theoretical studies in this area, therefore, have some value. Several quantum-dynamical calculations on the multiphoton excitation of SF6have been performed by wing the rotating-wave approximation (RWA).2-8 The most extensive and realistic calculations of this kind have been performed by Hodgkinson and co-workers.68 They use an accurate molecular Hamiltonian for the v3 vibrational mode which includes Coriolis and anharmonic octahedral splitting terms. They obtain encouraging agreement with experiment for the multiphoton excitation of the v3 vibrational mode of SF, at low temperatures. An important recent development in the theory of multiphoton excitation has been the application, by Leasure, Wyatt, and co-w~rkers,~J~ of Floquet theory'l to the vibrational-rotational excitation of diatomic molecules. This method enables exact multiphoton transition probabilities to be calculated from a given Hamiltonian. It requires the numerical solution of the time-dependent Schrodinger equation over the first optical cycle of the laser field, which is represented by the semiclassical electric (1)C. D. Cantrell, V. S. Letokhov, and A. A. Makarov in "Coherent Nonlinear Optics, Recent Advances", M. S. Feld and V. S. Letokhov, Eds., Springer-Verlag, West Berlin, 1980, p 165. (2) C. D. Cantrell and H. W. Galbraith, Opt. Commun., 21,374 (1977). (3) C. D. Cantrell and K. Fox, Opt. Lett., 2, 151 (1978). (4) J. R. Ackerhalt and H. W. Galbraith, J. Chem. Phys., 69, 1200 (1978). ' ( 5 ) H . W. Galbraith and J. R. Ackerhalt, Opt. Lett., 3, 109 (1978). (6) D. P. Hodgkinson, A. J. Taylor, and A. G. Robiette, J.Phys. B, 14, 1803 (1981). (7) A. J. Taylor, D. P. Hodgkinson, and A. G. Robiette, Opt. Commun., 41, 320 (1982). (8) D. P. Hodgkinson, A. J. Taylor, D. W. Wright, and A. G. Robiette, Chem. Phys. Lett., 90, 230 (1982). (9) S. C. Leasure, K. F. Milfeld, and R. E. Wvatt. J. Chem. Phvs.. 74. 6197 (1981). (10) S. Leasure and R. E. Wyatt, Opt. Eng.,19, 46 (1980). (11)J. H. Shirley, Phys. Reu., 138, B979 (1965). 0022-3654/03/2O07-O735$01.50/0
dipole approximation. We have recently used this technique in a study of molecular vibrational overtone transitions induced by intense infrared laser radiation.12 In the present work, the Floquet method is extended to the calculation of three-dimensional multiphoton excitation probabilities for the v3 vibrational mode of the SF, molecule. We also compare the Floquet theory with the RWA and show how the methods are related. The multiphoton transition probabilities for SF,, calculated with the Floquet method, are used to test the accuracy of the RWA for this system. We use the accurate vibration-rotation molecular Hamiltonian for the v3 vibrational mode of SF6previously employed by Hodgkinson and co-workem6 Section 2 gives brief summaries of the Floquet method and the RWA. A detailed discussion is also presented on how the two formalisms are related. The details of the SF, Hamiltonian are given in section 3. In section 4 the Floquet and RWA multiphoton calculations for the vg mode of SF6are compared. Conclusions are in section 5. 2(a). Floquet Method
The details of the application of Floquet theory to the laser-induced vibrational-rotational excitation of diatomic molecules have been presented by Leasure and co-workers? The extension of this method to octahedral molecules such as SF6is very straightforward and we will, therefore, only discuss the salient features of the method. The Hamiltonian for the molecular system interacting with the laser radiation, described by the semiclassical electric dipole approximation, is A = A0 - & t o cos (ut) (1) where A. is the Hamiltonian for the isolated diatomic molecule, pz is the component of the molecular dipole moment function projected along a space-fixed z axis, e,, is the classical electric field amplitude of the laser which is taken to be polarized along this z axis, and o is the frequency of the laser. The wave function solution to the time-dependent Schrodinger equation, for a given initial state at time t = 0, is expanded in the molecular eigenfunctions (Xkv(r)j (12) D. C. Clary, Mol. Phys., 46, 1099 (1982).
0 1903 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87, No. 5, 1983
+(r,t) = Caku(t) x d r )
Clary
(2)
kv
where r represents all the coordinates of the molecule. The notation ku denotes a molecular state with vibrational quantum number u and all other quantum numbers having the collective index k . The a k , ( t ) coefficients in eq 2 are obtained by solving the coupled set of equations ih [dak,(t)/dt] = Eokvakv(t)- Eo
COS ( U t )
Vkuk,u,UkJuJ(t)
k b'
(3)
The molecular eigenvalues are denoted by Eoku and Vkuk'"' is obtained by taking matrix elements over pz between the molecular states k u and k'u'. The evaluation of these matrix elements for SF6is discussed in section 3. The a k u ( t ) coefficients can be arranged as a column vector, there being one such vector for each initial state k'u'. By constructing a matrix with these column vectors, we have the time propagation matrix denoted by a(t). Diagonalization of a(t) at the end of the first optical cycle of time t = 2a/w yields the periodic Floquet eigensolutions 4(0). We have
4T(o) a(2a/w) 4(0) = exp(2aip)
d ih-bk,(t) dt
(5)
e0
= -2
vkuw"!bk!u,(t) exp[-i(Eok'u' k'v'
Eoku)t/h][eXp(id)+ eXp(-&)]
(11)
In the application of the RWA to a molecular problem involving a single vibrational mode, all the terms in eq 11 are omitted for which Iu - U'I # 1 (12) In the present Floquet and RWA calculations on SF6we only consider the v3 mode, and a dipole moment operator is used which only couples states for which 1u - u'I = 1. Thus, condition 12 is not an approximation in the present study. The approximation that is made in the present application of the RWA is to neglect all terms in eq 11 having the high-frequency exponents ( E o k l u - l - Eoku- nu) / h (13) (Eokfu+i - E o k u hw)/h The trial expansion bkv(t) = gkujeXp[i(Eokv/h- UW
(4)
where p is a diagonal matrix containing the Floquet characteristic exponents. We also have9
d t ) = a(t) d o ) exp(-iwt)
tions for the bku(t) coefficients
+ Aj)t]
(14)
is then made to yield a time-independent set of eigenvalue equations for the gkujand A; coefficients (UW
- Eoku/h)gkuj 4-
The Floquet eigensolutions are then related to the coefficients of eq 2 by a k u ( t ) = Cexp(ipmut)4kum(t)d'*k'u'm(O)
(6)
m
where m represents a Floquet state. The 4 k u m ( t ) functions can be expanded in the Fourier series
N
m
@ k u m ( t ) = @kum(O)
E C,kum exp(ijut)
I=-"
(7)
-
and the long-time average probability for the transition ku k'u'is obtained from Pkuk/u'
= C14k"m(0)1214k'u'm(0)12ckum m
There is an equation of this type for each ku state and j ranges from 1 to N , where N is the total number of molecular states. The most general solution of eq ll, in the RWA, will be a linear combination of the trial solutions
(8)
where
d;gku, eXp[i(Eoku/h - UW
bku(t) = j=1
5)t]
The coefficients d, are obtained from the initial conditions and can be different for each initial state. We are now in a position to relate the RWA and Floquet formalisms. Substitution of eq 16 into eq 10 and setting t = 2a/w gives +(r,2a/w) = CCdjgkuj exP[iAj2a/wIxkv(r) ku I
The numerical work required by the Floquet theory goes into (i) the solution of the coupled equations 3 over the first optical cycle, (ii) the diagonalization of the complex a matrix at the end of the first optical cycle, and (iii) the computation of the coefficients of eq 7. We have developed a procedurJ2 for solving the coupled equations 3 which is particularly efficient when large numbers of laser frequencies are considered since much of the numerical work can be done a t the first frequency. The technique uses the first-order Magnus appr~ximation.'~
ku
(17)
There is an equation of this type for each initial state and, from eq 17, it follows that the RWA approximation to the time-propagation matrix a(t) of section 2(a) at time t = 2a/u is a(2a/u) = g exp[ih2a/u]d (18) Here g has elements g,,,, where n represents the ku index, h is a diagonal matrix containing the A;, and d has elements d,,,, where n' labels the initial state K'u' and j labels an eigenvector of eq 15. Since we assume the initial condition G(r,O) = x k d r ) (19) it follows from eq 16 that d = gT. Comparison of eq 18 with the Floquet equation 4 shows
2(b). Rotating-Wave Approximation In the RWA, the wave function expansion
+(r,t) = x b k u ( t ) exp(-iEok,t/h)xku(r)
(16)
(10)
is used for a given initial state. Operation of eq 1 on eq 10 and expansion of cos ( u t )gives a set of coupled equa(13) P. Pechukas and J. C. Light, J. Chem. Phys., 44, 3897 (1966).
that these two equations are of the same form provided d(0) = g a n d X = p w . Thus, the RWA approximation to the Floquet eigensolution matrix 4(0) is obtained simply by determining the eigensolutions of eq 15. The RWA long-time average transition probabilities are then computed by using eq 8 with 4kum(0)replaced by gkum. From eq 5 and 16 it follows that the RWA approximation t o I $ k u m ( t ) is given by
The Journal of Physical Chemistty, Vol. 87, No. 5, 1983 737
Multiphoton Excitation of SF,
(20) and hence of eq 9 is equal to unity in the RWA. The accuracy of the RWA is dependent upon the validity of neglecting the terms with the high-frequency exponents of expression 13. This has been discussed in several previous works (see, for example, ref 14 and 15) and the approximation is valid provided I E o k + l - E o k u - h a [ , IEokTu+l - Eoku + hwl, I-(~o/2)Vkuk’u&ll 0, these are not good quantum numbers for molecules such as SF6. Inclusion of H1is essential for a realistic modeling of the multiphoton excitation of SFe6 The total number of basis functions that have to be included in a calculation is very large. Fortunately, symmetry can be used to factorize the Hamiltonian matrix into four separate blocks.6 All the parameters and equations
+
~~
~~
(14)M.Quack, J. Chem. Phys., 69,1282 (1978). (15) I. Schek, J. Jortner, and M. L. Sage, Chem. Phys., 59,ll (1981).
for determining the sF6molecular eigenfunctions are given in ref 6. An independent computer program was written to calculate these SF6 eigenfunctions. We obtained agreement with the eigenvalues of Hodgkinson et al?,16 to nine significant figures in every case tested. We also used the equations of Hodgkinson et a1.6 in calculating the integrals Vkuk’u’ over the dipole operator. These integrals couple together basis functions for which IV- V’I = 1, IL - L’I = 1,R = R’, and k R = kR’. In section 4 we report results for N(u,R’): the average number of photons absorbed for a u’-+ u transition, with the initial state u’ = 0, L’= 0, J’= R’, and kR’ = 0. These are defined by
and are summed over all angular momentum k states in the u level. In the results of section 4 we concentrate on two-photon transitions into the u = 2 state from the initial u’ = 0 state with the rotational quantum number R’ taking the values 1,2, and 3. Although results for many more initial R’states are required for a realistic comparison with experimental data, that is not the point of the present study in which we are simply comparing the Floquet and RWA methods. We report results for N(23’) at laser frequencies close to three different two-photon Q branch resonances, classified by the symmetry of the substates. These are for the Alg, E,, and F2gsubstates of the u = 2 level and occur close to the frequencies 944.5,945.8,and 948.3cm-l, respectively. There are sharp peaks in multiphoton spectra of SF6 at laser frequencies close to these three frequencies.” Identical basis sets were used in both the Floquet and RWA calculations. Most of the calculations in section 4 are for two-photon absorption. In these cases 33 basis functions were used to calculate the transition probabilities for initial R‘ = 2, with u, L, R, J, and kR taking the maximum values 2, 2, 6, 4,and 4,respectively. Results for initial R’ = 1and 3 are obtained in a completely separate calculation and in this case 31 basis functions were used with the same maximum quantum numbers as those for R’ = 2. Some limited three-photon absorption calculations are also presented in section 4 and these were done with a total of 63 basis functions, with u, L , R, J, and kR taking the maximum values 3, 3, 7, 4,and 4, respectively. With the SF6dipole moment function used in the calculations,6 the last RWA condition of eq 21 will only be violated for laser intensities larger than 88 TW cm-2. Thus, for practical laser intensities, the RWA conditions of eq 21 are satisfied for SF6. The RWA condition of eq 12 is met in the present calculations. However, we emphasize that there might be other problems for which the condition of eq 12 is not appropriate. 4. Results
The Floquet method requires the numerical solution of the time-dependent Schrodinger equation over the first optical cycle of the laser field, and results can be sensitive to the number of integration points (M)used in this procedure. Table I presents Floquet results of the average number of photons absorbed, N(2,R’),for a two-photon transition at the laser frequency and intensity of 944.5cm-’ and 10 MW cm-2, respectively. Results for M ranging from 16 to 100 are compared. It can be seen that the N(2,R’) (16) D. P. Hodgkinson, private communication. (17) S. S. Alimpiev, N. V. Karlov, S. M. Nikiforov, A. M. Prokhorov, B. G. Sartakov, E. M. Khokhlov, and A. L. Shtarkov, Opt. Commun., 31, 309 (1979).
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The Journal of Physical Chemistry, Vol. 87, No. 5, 1983
TABLE I: Average Two-Photon Absorption, N(2,R' ), as a Function of the Number of Integration Points (M) Used in Solving the Coupled Equationsa method
M
R'=1
R'=2
RI-3
Floquet
16 32 40 64 100
0.497 0.511 0.513 0.515 0.516 0.517
0.389 0.402 0.403 0.405 0.406 0.407
0.369 0.381 0.383 0.384 0.385 0.386
RWA
Laser intensity is 10 MW cm-z and laser frequency is 944.5 cm-'. a
TABLE 11: Average Two-Photon Absorption, N(2,R'), into Alp;SubstateP laser frequency/ cm944.48 944.49 944.50 944.51 944.52 944.53 944.54
R'= 1
R' = 3
0.162 0.163 0.275 0.277 0.513 0.517 0.902 0.904 0.890 0.888 0.502 0.502 0.269 0.270
0.115 0.116 0.201 0.203 0.403 0.407 0.838 0.841 0.884 0.882 0.436 0.436 0.214 0.214
0.104 0.105 0.184 0.186 0.383 0.386 0.839 0.842 0.852 0.850 0.390 0.391 0.187 0.188
TABLE 111: Average Two-Photon Absorption, N ( 2,R' ), into E, Substates"
945.75 945.80 945.85
R' = 1
N(2,R') R' = 2
R' = 3
0.0284 0.0287 0.499 0.502 0.0164 0.0166
0.0749 0.0759 0.152 0.154 0.0150 0.0152
0.107 0.109 0.0948 0.0956 0.0123 0.0125
a Laser intensity is 10 MW cm-'. First row contains Floquet results and second row contains RWA results.
TABLE IV: Average One- and Two-Photon Absorptions, N(1,R') and N(2,R'), into F,, Substatesa laser frequen-
laser fre-
quenCY
CY 1,
I
cm-I 948.0
TABLE V: Average Two-Photon Absorption, N(2,R'), into A,, Substatesa laser frequency/ cm-I 944.51 944.52 944.53
R' = 1
N(2,R' 1 R' = 2
R' = 3
0.0181 0.0182 0.563 0.567 0.0308 0.0310
0.0136 0.0137 0.644 0.648 0.0198 0.0199
0.0136 0.0137 0.970 0.972 0.0148 0.0149
a Laser intensity is 1 MW cm-z. First row contains Floquet results and second row contains RWA results.
TABLE VI: Average Two-Photon Absorption, N( 2,R'), into A,, Substatesa laser frequency/ cm-'
N(2,R') R' = 2
a Laser intensity is 10 MW cm-2. First row contains Floquet results and second row contains RWA results.
laser frequency/ cm-'
Clary
cmN ( 1 , R ' ) N(2,R') 0.508 0.0518 948.3 0.297 0.442 0.508 0.0522 0.298 0.440 948.1 0.440 0.136 948.4 0.205 0.120 0.440 0.137 0.206 0.121 948.2 0.352 0.614 948.5 0.158 0.0644 0.353 0.615 0.159 0.0647 a Laser intensity is 10 MW R' = 1. First row contains Floquet results and second row contains RWA results. N(1,R') N(2,R')
are converged to within 2% provided at least 32 integration points are used. In all the remaining calculations of this section, the minimum value of M used was 32. The RWA results for this laser frequency and intensity are also presented in Table I and these show excellent agreement
944.3 944.5 944.7
N( 2 3 ) 0.465 0.467 0.845 0.844 0.258 0.258
a Laser intensity is 100 MW R' = 1. First row contains Floquet results and second row contains RWA results.
TABLE VII: Average Number of Photons Absorbed for One-, Two-, and Three-Photon Transitionsa Floquet RWA
0.0237 0.0237
0.001 08 0.001 08
0.140 0.139
a Laser intensity is 1 0 MW cm-l and frequency is 946.38 cm-'; R' = 1.
with the converged Floquet results. Tables 11-IV display comparisons of the average twophoton absorptions, N(2,R'), calculated by using the Floquet and RWA methods, for the three different laser frequency ranges corresponding to the two-photon resonance transitions into the Alg, E , and F,, substates. Average one-photon absorptions, N?l,R'), are also presented for the F,, substates in Table IV as these are particularly large for laser frequencies close to 948.0 cm-'. In all the calculations of Tables 11-IV,the laser intensity was 10 MW cm-2 as this has been the intensity used in previous multiphoton calculations in which detailed comparisons with experiment have been made.6 All the Floquet and RWA results of Tables 11-IV are, once again, in excellent agreement. It is of interest to examine how the results vary with laser intensity. Tables V and VI present Floquet and RWA calculations of N(2,R') for the Al, substates with laser intensities of 1 and 100 MW cm-2, respectively. Excellent agreement between these Floquet and RWA results is obtained. Comparisons of the results of Tables V and VI give a good illustration as to how sensitive the N(2,R') are to laser intensity. Our calculations have concentrated on the two-photon spectra. The three-photon resonance spectra have very narrow bandwidths and are very difficult to detect for low values of R! An example of the average number of photons absorbed (N(3,l))for a three-photon resonance transition is given in Table VII, in which results for N(1,l) and N(2,l) are also presented, for comparison, at the laser frequency of 946.38 cm-' and laser intensity of 10 MW cm-2. This laser frequency corresponds to a three-photon transition into the set of substates with Flu symmetry. The agreement between the Floquet and RWA results of Table VII,
J. Phys. Chem. 1983, 87, 739-741
is, once again, almost perfect. We find that, in every case, which we have considered, the RWA results are in excellent agreement with the “exact” Floquet results. As is discussed in section 2, the RWA is expected to work well for SF6,but what might not have been expected is just how accurate the RWA is for this system. From the computational viewpoint, the RWA requires the construction and diagonalization of a real matrix (see eq 15) to obtain the multiphoton transition probabilities while the Floquet technique needs the diagonalization of the complex matrix in eq 4 of the same dimension. Furthermore, the Floquet method requires the numerical solution of the time-dependent Schrodinger equation over the first optical cycle of the laser field, although this can be made very efficient when results for a large number of laser frequencies are required.12 The computation of the CPmcoefficients of eq 7 can also be expensive in disk storage and retrieval time. However, we have found in the Floquet calculations on SF6that, if Ckumis set to unity, the transition probabilities remain unchanged to three significant figures. Despite this, in all the calculations of Tables I-VII, the Cjko”’ coefficients were computed. The diagonalization of the complex matrix is the main feature that makes the Floquet method typically 1order of magnitude more expensive in computer time than the RWA. Furthermore, the Floquet technique requires much more core storage space than the RWA, and, for these reasons, we are only able to use a maximum of 80 basis functions in our Floquet calculations on the CDC 7600 computer while RWA computations with more than 200 basis functions are feasible. We are, therefore, unable to
739
perform extensive Floquet calculations with total angular momentum quantum numbers larger than 5 and these computations are required to enable a comparison with experimental data to be carried out at realistic temperatures. RWA calculations on SF6for quite large values of J can be performed, however.6-s 5. Conclusions The Floquet method and the rotating-wave approximation for calculating multiphoton transition probabilities have been compared and the connection between the techniques has been discussed. A new method for correcting the RWA has also been proposed. Calculations on the multiphoton absorption in the v3 vibrational mode of SF6have been carried out by using the Floquet and RWA methods. A realistic molecular Hamiltonian has been used in the calculations and vibrational and rotational angular momentum coupling is treated accurately. The Floquet and RWA calculations of the average number of photons absorbed in SF6 are in excellent agreement in every case studied. The RWA requires much less computer time and core storage than the Floquet method. Our findings suggest that complete confidence can be had in the accuracy of RWA multiphoton calculations on SF6and we would expect this accuracy to hold for other, similar molecules.
Acknowledgment. I am grateful to Dr. D. P. Hodgkinson for forwarding calculated values of the SF6 molecular eigenvalues, and also for useful discussions. Registry No. SF6,2551-62-4.
Structural Change of a Surface-Active Azo Dye on Adsorption at the Aqueous Solutlon-Carbon Tetrachloride Interface Studled by Resonance Raman Spectra Hlromlchl Takahashi,+ Junzo Umemura, and Tohru Takenaka Institute for Chemlcal Research, Kyoto University, Ujl, Kyoto-Fu, 6 1 1, Japan (Received: August 23, 1982; I n Final Form: October 20, 1982)
Resonance Raman spectra of adsorbed monolayers of a surface-active azo dye, brilliant red BS (BRBS),at the water-carbon tetrachloride interface were measured by the total reflection method. It was found that a structural change of BRBS from the trivalent azo form to the divalent hydrazone form took place on adsorption from a basic aqueous solution below the cmc. This structural change was the same as that previously observed on micellization of BRBS in a basic solution. The adsorption isotherm of BRBS obtained from Raman intensity measurements was in good agreement with the one calculated from the interfacial tension-concentration curve by the Gibbs adsorption equation.
Introduction It is well-known that there are two characteristic behaviors of surfactants in aqueous solutions. One is micelle formation above the critical micelle concentration (cmc) and the other is adsorption and orientation at the interface between the aqueous solution and air or oil. Although these have been of the greatest interest for long time in surface chemistry, molecular spectroscopic studies have been very limited in number.
In a preceding paper,l we have measured the resonance Raman spectra of aqueous solutions of a surface-active azo dye, brilliant red BS (BRBS), over a wide range of concentrations in neutral and basic media. It was found for the neutral solution that the divalent hydrazone form (Ib in Figure 1)was the predominant species in a tautomeric mixture of Ia and Ib both below and above the cmc. In basic solution, however, the deprotonated trivalent azo form (IIa) was a major contributor to a resonance hybrid
Present address: Tochigi Research Laboratory, Kao Soap Co. Ltd., Ichikai-Machi, Haga, Tochigi, 321-34, Japan.
(1) H. Takahashi, J. Umemura, and T. Takenaka, J.Phys. Chem., 86, 4660 (1982).
0022-3654/83/2087-0739$01.50/00 1983 American Chemical Society
