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Ab initio calculations of doubly resonant sum-frequency generation second-order polarizabilities of LiH
Chemical Physics Letters 380 (2003) 549–555 www.elsevier.com/locate/cplett
Ab initio calculations of doubly resonant sum-frequency generation second-order polarizabilities of LiH Robert Zalesny
a,b,c
, Wojciech Bartkowiak a, Beno^ıt Champagne
c,*
a
b
Institute of Physical and Theoretical Chemistry, Wroclaw University of Technology, Wybrze_ze Wyspia nskiego 27, 50-370 Wrocław, Poland Faculty of Chemistry, Department of Quantum Chemistry, Nicolaus Copernicus University, Gagarina 7, PL-87 100 Toru n, Poland c Laboratoire de Chimie Th eorique Appliqu ee, Facult es Universitaires Notre-Dame de la Paix, rue de Bruxelles 61, B-5000 Namur, Belgium Received 7 August 2003; in final form 6 September 2003 Published online: 7 October 2003
Abstract In the present Letter doubly resonant sum-frequency generation (DR-SFG) second-order polarizabilities (b) are reported for the lithium hydride molecule. The calculations, which refer to the diagonal longitudinal component of b and ignore rotation, are based on data evaluated using the multireference configuration interaction method with singles and doubles. The DR-SFG results are partitioned into their infrared–visible and visible–infrared components. They are analyzed as a function of the infrared and visible frequencies as well as the number of vibrational states considered in the sum-over-states expressions. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Recently, the sum-frequency generation spectroscopy (also referred to as SFG) has attracted significant interest because of its applications in surface science [1–3]. In SFG technique one employs two different laser beams of different frequencies. These are usually chosen to be in infrared (xir ) and visible (xvis ) ranges. In the onecolour variant of the SFG spectroscopy one of the
*
Corresponding author. Fax: +3281724567. E-mail address: [email protected] (B. Champagne).
beams is far from resonance while the second is scanning certain spectral range. The doubly resonant sum-frequency generation spectroscopy (DR-SFG) is also known as two-colour SFG spectroscopy. In this technique, both beams are chosen to be near the resonance with electronic and/or vibrational transitions (see Fig. 1). Using DR-SFG allows to study electron-vibration coupling in molecules [4]. Although the general theory describing DRSFG processes has been developed recently by Huang and Shen [5] and Lin et al. [6,7] we are not aware of any first-principles calculations related to real systems. The lithium hydride molecule has been chosen for this first application. Indeed, on
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.09.042
550
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
ω ir
|lE>
babc ðxr ; x1 ; x2 Þ ^b jL;lihl; Lj^ la jK;kihk; Kjl lc j0;0i 1 X 0 X 0 h0; 0j^ ¼ 2 ðxkK xr ÞðxlL x2 Þ h kK lL
ω vis
|l0>
^c jL; lihl;Lj^ la jK; kihk; Kjl lb j0;0i 1 X 0 X 0 h0;0j^ ; 2 ðxkK xr ÞðxlL x1 Þ h kK lL
|00>
ð2Þ
ω vis
þ
|00>
ω ir
Fig. 1. Schematic representation of two dominant second-order nonlinear optical processes contributing to DR-SFG.
the one hand, the potential energy and dipole transition curves, which enter the evaluation of the DR-SFG b, have been calculated by several authors and are in excellent agreement with experimental data [8–14]. On the other hand, numerous studies reported on the results of calculations of the molecular dipole polarizabilities up to third order with the inclusion of vibrational contributions [15–22].
where K and L label the electronic states involved in the process (in particular they can refer to the same state). Obviously, for the (xir ; xvis ) combination of frequencies the longitudinal second-order response reads: ir–vis þ bvis–ir bzzz ðxr ; x1 ; x2 Þ ¼ bzzz zzz
ð3Þ
with vis–ir bzzz
¼
^z jE;lihl;Ej^ 1 X 0 X 0 h0;0j^ lz jE;kihk;Ejl lz j0;0i ; 2 ðx x x Þðx x h k kE vis ir lE vis Þ l ð4Þ
and bir–vis zzz
2. Computational aspects
¼ The sum-over-states expression for second-order polarizability (b) reads [23,24]: babc ðxr ; x1 ; x2 Þ 2
¼ h P ða; b; c; xr ; x1 ; x2 Þ X 0 X 0 h0; 0j^ ^b jL; lihl; Lj^ la jK; kihk; Kjl lc j0; 0i ; x Þðx x Þ ðx kK r lL 2 kK lL ð1Þ
^ is the fluctuation dipole moment operator where l ^ ¼ l^ h0; 0j^ l lj0; 0i and xr ¼ x1 þ x2 . P stands for the permutation operator. In labelling different vibrational and electronic states we follow the notation of Bishop [23]. Small letters refer to vibrational states while the capital letters denote electronic states. Then, hxkK is the energy of the kth vibrational state of Kth electronic state relative to the ground state (0,0). For the doubly resonant case (x1 þ x2 ’ xkK ; x1 or x2 ’ xlL ) the off-resonant terms can be neglected and one arrives at
^z j0;lihl;0j^ 1 X 0 X 0 h0;0j^ lz jE;kihk;Ejl lz j0;0i ; 2 ðx x x Þðx x h k kE ir vis l0 ir Þ l ð5Þ
where each term is also represented in Fig. 1. E labels the electronic excited state in resonance with the excitation source. Such partitioning into bir–vis and bvis–ir turns out to be convenient for analysis purpose [7]. Once the dipole moment, the transition dipole functions as well as the vibrational eigenfunctions for the two (ground and excited) states are known, both Eqs. (4) and (5) can be evaluated. In the present study, the potential energy and transition dipole curves for the ground X 1 Rþ and excited A1 R states were taken from the work of Partridge and Langhoff [11]. They used multireference configuration interaction method with singles and doubles (MR-SDCI) with a Slater basis set consisting of 22r, 12p and 7d orbitals. We used these data in order to solve the radial Schr€ odinger equation for the vibrational problem using Numerov–Cooley method as implemented in MO L C A S series of programs [25]. All computa-
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
tions of the longitudinal component of b reported herein ignore rotation (J ¼ 0).
3. Results and discussion In Table 1, the comparison between calculated and experimental spacings between vibrational levels is presented [26]. The quantity Gm is defined as Gm ¼ Eðm; 0Þ Eð0; 0Þ;
m ¼ 0; 1; 2; . . . ;
ð6Þ
where Eðm; J Þ is an energy for vibrational level with vibrational and rotational quantum numbers m and J . The values of DGm are presented only for the four lowest vibrational states because the enhancement of b shall be investigated when approaching the lowest infrared resonances in both electronic states. It appears that the anharmonic nature of the potential energy curve and its influence on the vibrational transition frequencies are well reproduced by our approach. The energy
551
difference between the jE; 0i and j0; 0i vibronic states calculated from the data reported by Partridge et al. is equal to 25650 cm1 (0.1169 a.u.) [10]. The expressions defining both contributions to the second-order polarizability imply summation over the vibrational levels of the X 1 Rþ and A1 R electronic states. Hence, the problem of convergence of these expansions immediately arises. In Fig. 2 we present the results of calculations of the longitudinal component of the second-order polarizability (bzzz ) as a function of the number of terms included in the k- and l-summations of Eqs. (4) and (5). Two cases are considered. In the first case the same number of vibrational levels was included for both electronic states (black squares). As it is clearly seen both expansions are well converged with 20 states. In all subsequent calculations we include 25 vibrational levels for both electronic states. However, it is interesting to consider the second case when the number of levels in A1 R state was fixed and set equal to 25 (black
Table 1 Calculated and experimental values of spacings between the four lowest vibrational levels, DGm ¼ Gmþ1 Gm X 1 Rþ
A1 R
m
DGcalc m
DGexpt m
m
DGcalc m
DGexpt m
0 1 2 3
1353.5 1308.7 1264.9 1221.8
1359.8 1314.8 1270.9 1227.8
0 1 2 3
295.5 325.5 347.0 362.9
280.8 313.0 335.7 352.8
The values are given in cm1 . Experimental values are taken from [26].
0
0
-50
-5000 βzzz [au]
βzzz [au]
-100 -150 -200
-10000 -15000
-250 -20000
-300 -350
-25000 0
5
10
15
20
25
0
5
10
15
20
25
Fig. 2. Dependence of the second-order polarizability on the number of vibrational states included in the summation. The squares (j) denote the case when the same number of levels were taken for both states while the triangles (N) label the situation where the number of vibrational levels in the A1 R state is set equal to 25. xvis ¼ 0:110 and xir ¼ 0:005 a.u.
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R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
triangles on the plots). Since the potential energy curve for X 1 Rþ state in good approximation is harmonic, one should expect very fast convergence in the ir–vis case. As it is seen from Fig. 2, the convergence is reached after inclusion of two excited vibrational levels. In the vis–ir case there is no summation over the vibrational states of the ground electronic state, so the b values are not sensitive to the vibrational structure in the X 1 Rþ state. ir–vis The dependence of bzzz on infrared frequency ranging from xir ¼ 0:0015 a.u. (329 cm1 ) to xir ¼ 0:0060 a.u. (1317 cm1 ) is presented in Table 2. It can be seen that near the latter limiting value, the bzzz is strongly enhanced. The inspection of Eq. (5) shows that the denominator becomes very small when hxir is close to the energy spacing between the two first vibrational levels of the electronic ground state ( hx0l ). In the case of X 1 Rþ electronic state, the energy gap between m ¼ 1 and m ¼ 0 vibrational levels is 1353 cm1 . Hence, when increasing the infrared angular frequency the (x0l xir ) term becomes small and the value of ir–vis bzzz increases substantially. The dependence of bzzz on the visible frequency is presented in Table 3. Obviously, the increase of the visible frequency leads to an increase of the bzzz values. Since the energy difference between ground vibrational states of X 1 Rþ and A1 R is 0.1169 a.u., the values of xvis were chosen to be far from the electronic resonance and approach it gradually. Analogously to the infrared-visible term, the key
Table 3 bir–vis and bvis–ir values as a function of the visible frequency zzz zzz xvis
bir–vis zzz
bvis–ir zzz
0.080 0.085 0.090 0.092 0.096 0.100 0.105 0.110
)7.4 )8.5 )10.1 )10.9 )12.7 )15.4 )20.3 )29.6
)2009 )2514 )3241 )3626 )4639 )6165 )9507 )17 015
xir ¼ 0:001 a.u. All values are in a.u. vis–ir with the factor responsible for the increase of bzzz frequencies is the decrease of the denominator value. The comparison of the absolute values of both bzzz contributions brings about the problem of explaining the observed large differences, i.e. the bvis–ir values are usually larger than their bir–vis analogs. In order to explain this difference, we partitioned each of the expansions defined by Eqs. (4) and (5) into two components (Table 4). The first one comprises the diagonal contributions denoted as kk (k ¼ l) while the second one comprised the off-diagonal contributions kl (k 6¼ l). In the infrared-visible case, the off-diagonal part of b is significantly larger than the diagonal contribution. On the other hand, a totally different pattern is observed for the visible-infrared term where both kk and kl contributions are of similar magnitude. The diagonal term of bvis–ir reads vis–ir bzzz ¼
Table 2 vis–ir bir–vis and bzzz values as a function of the infrared frequency zzz xir 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060
(329.2) (438.9) (548.7) (658.4) (768.2) (877.9) (987.6) (1097.4) (1207.1) (1316.8)
bir–vis zzz
bvis–ir zzz
)35 )42 )51 )65 )84 )114 )167 )271 )550 )2607
)17 662 )18 366 )19 137 )19 987 )20 930 )21 987 )23 181 )24 553 )26 155 )28 077
xvis ¼ 0:110 a.u. All values are in a.u. except those in parentheses which are given in cm1 .
0 1 X DlEk;00 h2 k
h0; 0j^ lz jE; kihk; Ej^ lz j0; 0i ; ðxkE xvis xir ÞðxkE xvis Þ
ð7Þ
where DlEk;00 stands for the z-component of the dipole moment difference between the kth vibra-
Table 4 Partitioning of the b values into their diagonal (kk) and off-diagonal (kl) terms
bir–vis zzz bvis–ir zzz
kk
kl
Total
)20 )10 816
)251 )13 737
)271 )24 553
xvis ¼ 0:110 and xir ¼ 0.005 a.u. All values are in a.u.
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
tional state of the electronic excited state E and the ground electronic state with m ¼ 0. This diagonal contribution is of the same order of magnitude as
[a.u.]
ωvis = 0.110 a.u. 2.0e+05
vis-ir
8.0e+04
β
β
its off-diagonal counterpart that is proportional to hk; Ej^ lz jE; li, of which the leading harmonic part is proportional to the derivative of the E-excited
ωvis = 0.110 a.u.
ir-vis
[a.u.]
1.2e+05
4.0e+04
0 0.001
0.003
0.006
0.009
1.0e+05
0.001
infrared frequency [a.u.]
[a.u.]
9.0e+05
0.006
0.009
ωvis = 0.115 a.u.
6.0e+05
β
vis-ir
3.0e+07
0.003
infrared frequency [a.u.]
ωvis = 0.115 a.u.
ir-vis
[a.u.]
4.5e+07
β
553
1.5e+07
0 0.001
3.0e+05
0 0.001
0.003 0.006 0.009 infrared frequency [a.u.]
1.2e+07
ωvis = 0.120 a.u.
0.003 0.006 0.009 infrared frequency [a.u.]
ωvis = 0.120 a.u.
[a.u.] β
2.0e+06 0 0.001
8.0e+06
vis-ir
4.0e+06
β
ir-vis
[a.u.]
6.0e+06
0.003
0.006
4.0e+06
0 0.001
0.009
infrared frequency [a.u.]
[a.u.] vis-ir
β
β
4.0e+05
0.003
0.006
0.009
ωvis = 0.125 a.u.
8.0e+05
0 0.001
0.006
infrared frequency [a.u.]
ωvis = 0.125 a.u.
ir-vis
[a.u.]
1.2e+06
0.003
0.009
8.0e+06
4.0e+06
0 0.001
0.003
0.006
Fig. 3. Dispersion of bir–vis and bvis–ir as a function of xir for different values of xvis . zzz zzz
0.009
554
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
state dipole moment with respect to the stretching vibrational normal coordinate. On the other hand, hl; 0j^ lz j0; 0i is the component of the diagonal and off-diagonal bir–vis numerators which differ with respect to the bvis–ir numerator. Its value, which is about two orders of magnitude smaller than hk; Ej^ lz jE; li, accounts largely for the difference of magnitude between bir–vis and bvis–ir . The remaining factors which influence the relative magnitudes of bir–vis and bvis–ir are related to overlaps between vibrational wavefunctions as well as to mechanical and electrical anharmonicities. In order to evaluate b under resonance it was compulsory to include in the treatment a damping factor (C). A value of 10 cm1 has been chosen for all vibronic states. In Fig. 3 we present the real part of the two dominant components of DR-SFG b: Re bir–vis 1 X0X0 ¼ 2 h k l " h0; 0j^ lz jE; kihk; Ejl^z j0; lihl; 0j^ lz j0; 0i 2
2
½ðxkE xr Þ þ 14 C2kE ½ðxl0 xir Þ þ 14 C2l0 # 1 ½ðxkE xr Þðxl0 xir Þ CkE Cl0 ð8Þ 4 Re bvis–ir 1 X0X0 h2 k l " h0; 0j^ lz jE; kihk; Ejl^z jE; lihE; lj^ lz j0; 0i
spectra as well as for different values of xvis . On the other hand, for the vis–ir process, the resonances with the successive vibrational levels of the excited electronic state are clearly visible. For both processes, changing the xvis frequency leads to variations of b that can attain several orders of magnitude.
4. Summary and outlook DR-SFG second-order polarizabilities have been reported for the lithium hydride molecule. The calculations refer to the diagonal longitudinal component of b, ignore rotation, and are based on data calculated using the multireference configuration interaction method with singles and doubles. The convergence with respect to the number of vibrational states as well as the dependence upon both infrared and visible frequencies have been addressed. The total DR-SFG response has been partitioned into its infrared–visible and visible–infrared dominant components and it has been shown that the later is the largest. This has been related to the similar magnitude of the diagonal (kk) and off-diagonal (kl) contributions to the visible–infrared term. Extension to larger systems, including conjugated organic compounds, of such ab initio treatments to DR-SFG is expected to be a useful tool to interpret the physics behind complex DR-SFG spectra [4].
¼
2
½ðxkE xr Þ þ
1 2 C ½ðxlE 4 kE
2
xvis Þ þ
Acknowledgements
1 2 C 4 lE
#
1 ½ðxkE xr ÞðxlE xvis Þ CkE ClE : 4
ð9Þ
For xvis values ranging from 0.110 to 0.125 a.u., the bir–vis spectra display a resonance at 0.0062 a.u. which corresponds to the m ¼ 0 m ¼ 1 vibrational transition of the ground electronic state. In addition, there are bands separated by approximately 0.0015 a.u. that correspond to resonances with successive vibrational levels of the excited electronic state. However, the intensities of these transitions can differ strongly within the same
The authors thank Professor Andrzej J. Sadlej for his help and advices. One of the authors (RZ) gratefully acknowledges the financial support from CGRI (Commissariat General aux Relations Internationales) during his stay in Namur. BC thanks the Belgian National Fund for Scientific Research for his Senior Research Associate position. We acknowledge the support from the Interuniversity Attraction Poles Programme on ÔSupramolecular Chemistry and Supramolecular Catalysis (IUAP no. P5-03)Õ–Belgian State-Federal Office for Scientific, Technical and Cultural Affairs. This work was facilitated in part by Wroclaw
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
University of Technology. We also acknowledge the use of the Sun Enterprise 6000 at the University Center for Information Technology of the Nicolaus Copernicus University. Part of the calculations have also been performed on the Pentium III/Pentium IV cluster of the CTA lab for which the authors acknowledge the financial support of the FNRS. References [1] Z. Chen, Y.R. Shen, G.A. Somorjai, Annu. Rev. Phys. Chem. 53 (2002) 437. [2] S. Baldelli, N. Markovic, P. Ross, Y.R. Shen, G.A. Somorjai, J. Phys. Chem. B 103 (1999) 8920. [3] X. Su, P.S. Cremer, Y.R. Shen, G.A. Somorjai, J. Am. Chem. Soc. 119 (1997) 3994. [4] C. Humbert, L. Dreesen, S. Nihonyanagi, T. Masuda, T. Kondo, A.A. Mani, K. Uosaki, P.A. Thiry, A. Peremans, Appl. Surf. Sci. 212–213 (2003) 797. [5] J.Y. Huang, Y.R. Shen, Phys. Rev. A 49 (1994) 3973. [6] S.H. Lin, M. Hayashi, R. Islampour, J. Yu, D.Y. Yang, G.Y.C. Wu, Physica B 222 (1996) 191. [7] M. Hayashi, S.H. Lin, M.B. Raschke, Y.R. Shen, J. Phys. Chem. A 106 (2002) 2271. [8] K.K. Docken, J. Hinze, J. Chem. Phys. 57 (1972) 4928.
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[9] K.K. Docken, J. Hinze, J. Chem. Phys. 57 (1972) 4936. [10] H. Partridge, S.R. Langhoff, W.C. Stwalley, W.T. Zemnke, J. Chem. Phys. 75 (1981) 2299. [11] H. Partridge, S.R. Langhoff, J. Chem. Phys. 74 (1981) 2361. [12] B. J€ onsson, B.O. Roos, P.R. Taylor, P.E.M. Siegbahn, J. Chem. Phys. 74 (1981) 4566. [13] A. Boutalib, F.X. Gadea, J. Chem. Phys. 97 (1992) 1144. [14] X. Li, J. Paldus, J. Chem. Phys. 118 (2003) 2470. [15] J.M. OÕHare, R.P. Hurst, J. Chem. Phys. 46 (1967) 2356. [16] M.G. Papadopoulos, A. Willetts, N.C. Handy, A.E. Underhill, Mol. Phys. 88 (1996) 1063. gren, J. Chem. Phys. 105 [17] D. Jonsson, P. Norman, H. A (1996) 6401. [18] B.O. Roos, A.J. Sadlej, J. Chem. Phys. 76 (1982) 5444. [19] M. Rerat, M. Merawa, C. Pouchan, Phys. Rev. A 46 (1992) 5471. [20] J. Vrbik, D.A. Legare, S.M. Rothstein, J. Chem. Phys. 92 (1990) 1221. [21] D.M. Bishop, B. Lam, Chem. Phys. Lett. 120 (1985) 69. [22] M. Cafiero, L. Adamowicz, M. Duran, J.M. Luis, J. Mol. Struct. (THEOCHEM) 633 (2003) 113. [23] D.M. Bishop, Adv. Chem. Phys. 104 (1998) 1. [24] B. Kirtman, B. Champagne, Int. Rev. Phys. Chem. 16 (1997) 389. [25] K. Andersson, et al., MOLCAS VERSION 5.2, Dept. of Theor. Chem., Chem. Center, University of Lund, P.O.B. 124, 2-221 00 Lund, Sweden, Lund, 2000. [26] F.B. Orth, W.C. Stwalley, J. Mol. Spectrosc. 17 (1979) 76.
Ab initio calculations of doubly resonant sum-frequency generation second-order polarizabilities of LiH Robert Zalesny
a,b,c
, Wojciech Bartkowiak a, Beno^ıt Champagne
c,*
a
b
Institute of Physical and Theoretical Chemistry, Wroclaw University of Technology, Wybrze_ze Wyspia nskiego 27, 50-370 Wrocław, Poland Faculty of Chemistry, Department of Quantum Chemistry, Nicolaus Copernicus University, Gagarina 7, PL-87 100 Toru n, Poland c Laboratoire de Chimie Th eorique Appliqu ee, Facult es Universitaires Notre-Dame de la Paix, rue de Bruxelles 61, B-5000 Namur, Belgium Received 7 August 2003; in final form 6 September 2003 Published online: 7 October 2003
Abstract In the present Letter doubly resonant sum-frequency generation (DR-SFG) second-order polarizabilities (b) are reported for the lithium hydride molecule. The calculations, which refer to the diagonal longitudinal component of b and ignore rotation, are based on data evaluated using the multireference configuration interaction method with singles and doubles. The DR-SFG results are partitioned into their infrared–visible and visible–infrared components. They are analyzed as a function of the infrared and visible frequencies as well as the number of vibrational states considered in the sum-over-states expressions. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Recently, the sum-frequency generation spectroscopy (also referred to as SFG) has attracted significant interest because of its applications in surface science [1–3]. In SFG technique one employs two different laser beams of different frequencies. These are usually chosen to be in infrared (xir ) and visible (xvis ) ranges. In the onecolour variant of the SFG spectroscopy one of the
*
Corresponding author. Fax: +3281724567. E-mail address: [email protected] (B. Champagne).
beams is far from resonance while the second is scanning certain spectral range. The doubly resonant sum-frequency generation spectroscopy (DR-SFG) is also known as two-colour SFG spectroscopy. In this technique, both beams are chosen to be near the resonance with electronic and/or vibrational transitions (see Fig. 1). Using DR-SFG allows to study electron-vibration coupling in molecules [4]. Although the general theory describing DRSFG processes has been developed recently by Huang and Shen [5] and Lin et al. [6,7] we are not aware of any first-principles calculations related to real systems. The lithium hydride molecule has been chosen for this first application. Indeed, on
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.09.042
550
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
ω ir
|lE>
babc ðxr ; x1 ; x2 Þ ^b jL;lihl; Lj^ la jK;kihk; Kjl lc j0;0i 1 X 0 X 0 h0; 0j^ ¼ 2 ðxkK xr ÞðxlL x2 Þ h kK lL
ω vis
|l0>
^c jL; lihl;Lj^ la jK; kihk; Kjl lb j0;0i 1 X 0 X 0 h0;0j^ ; 2 ðxkK xr ÞðxlL x1 Þ h kK lL
|00>
ð2Þ
ω vis
þ
|00>
ω ir
Fig. 1. Schematic representation of two dominant second-order nonlinear optical processes contributing to DR-SFG.
the one hand, the potential energy and dipole transition curves, which enter the evaluation of the DR-SFG b, have been calculated by several authors and are in excellent agreement with experimental data [8–14]. On the other hand, numerous studies reported on the results of calculations of the molecular dipole polarizabilities up to third order with the inclusion of vibrational contributions [15–22].
where K and L label the electronic states involved in the process (in particular they can refer to the same state). Obviously, for the (xir ; xvis ) combination of frequencies the longitudinal second-order response reads: ir–vis þ bvis–ir bzzz ðxr ; x1 ; x2 Þ ¼ bzzz zzz
ð3Þ
with vis–ir bzzz
¼
^z jE;lihl;Ej^ 1 X 0 X 0 h0;0j^ lz jE;kihk;Ejl lz j0;0i ; 2 ðx x x Þðx x h k kE vis ir lE vis Þ l ð4Þ
and bir–vis zzz
2. Computational aspects
¼ The sum-over-states expression for second-order polarizability (b) reads [23,24]: babc ðxr ; x1 ; x2 Þ 2
¼ h P ða; b; c; xr ; x1 ; x2 Þ X 0 X 0 h0; 0j^ ^b jL; lihl; Lj^ la jK; kihk; Kjl lc j0; 0i ; x Þðx x Þ ðx kK r lL 2 kK lL ð1Þ
^ is the fluctuation dipole moment operator where l ^ ¼ l^ h0; 0j^ l lj0; 0i and xr ¼ x1 þ x2 . P stands for the permutation operator. In labelling different vibrational and electronic states we follow the notation of Bishop [23]. Small letters refer to vibrational states while the capital letters denote electronic states. Then, hxkK is the energy of the kth vibrational state of Kth electronic state relative to the ground state (0,0). For the doubly resonant case (x1 þ x2 ’ xkK ; x1 or x2 ’ xlL ) the off-resonant terms can be neglected and one arrives at
^z j0;lihl;0j^ 1 X 0 X 0 h0;0j^ lz jE;kihk;Ejl lz j0;0i ; 2 ðx x x Þðx x h k kE ir vis l0 ir Þ l ð5Þ
where each term is also represented in Fig. 1. E labels the electronic excited state in resonance with the excitation source. Such partitioning into bir–vis and bvis–ir turns out to be convenient for analysis purpose [7]. Once the dipole moment, the transition dipole functions as well as the vibrational eigenfunctions for the two (ground and excited) states are known, both Eqs. (4) and (5) can be evaluated. In the present study, the potential energy and transition dipole curves for the ground X 1 Rþ and excited A1 R states were taken from the work of Partridge and Langhoff [11]. They used multireference configuration interaction method with singles and doubles (MR-SDCI) with a Slater basis set consisting of 22r, 12p and 7d orbitals. We used these data in order to solve the radial Schr€ odinger equation for the vibrational problem using Numerov–Cooley method as implemented in MO L C A S series of programs [25]. All computa-
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
tions of the longitudinal component of b reported herein ignore rotation (J ¼ 0).
3. Results and discussion In Table 1, the comparison between calculated and experimental spacings between vibrational levels is presented [26]. The quantity Gm is defined as Gm ¼ Eðm; 0Þ Eð0; 0Þ;
m ¼ 0; 1; 2; . . . ;
ð6Þ
where Eðm; J Þ is an energy for vibrational level with vibrational and rotational quantum numbers m and J . The values of DGm are presented only for the four lowest vibrational states because the enhancement of b shall be investigated when approaching the lowest infrared resonances in both electronic states. It appears that the anharmonic nature of the potential energy curve and its influence on the vibrational transition frequencies are well reproduced by our approach. The energy
551
difference between the jE; 0i and j0; 0i vibronic states calculated from the data reported by Partridge et al. is equal to 25650 cm1 (0.1169 a.u.) [10]. The expressions defining both contributions to the second-order polarizability imply summation over the vibrational levels of the X 1 Rþ and A1 R electronic states. Hence, the problem of convergence of these expansions immediately arises. In Fig. 2 we present the results of calculations of the longitudinal component of the second-order polarizability (bzzz ) as a function of the number of terms included in the k- and l-summations of Eqs. (4) and (5). Two cases are considered. In the first case the same number of vibrational levels was included for both electronic states (black squares). As it is clearly seen both expansions are well converged with 20 states. In all subsequent calculations we include 25 vibrational levels for both electronic states. However, it is interesting to consider the second case when the number of levels in A1 R state was fixed and set equal to 25 (black
Table 1 Calculated and experimental values of spacings between the four lowest vibrational levels, DGm ¼ Gmþ1 Gm X 1 Rþ
A1 R
m
DGcalc m
DGexpt m
m
DGcalc m
DGexpt m
0 1 2 3
1353.5 1308.7 1264.9 1221.8
1359.8 1314.8 1270.9 1227.8
0 1 2 3
295.5 325.5 347.0 362.9
280.8 313.0 335.7 352.8
The values are given in cm1 . Experimental values are taken from [26].
0
0
-50
-5000 βzzz [au]
βzzz [au]
-100 -150 -200
-10000 -15000
-250 -20000
-300 -350
-25000 0
5
10
15
20
25
0
5
10
15
20
25
Fig. 2. Dependence of the second-order polarizability on the number of vibrational states included in the summation. The squares (j) denote the case when the same number of levels were taken for both states while the triangles (N) label the situation where the number of vibrational levels in the A1 R state is set equal to 25. xvis ¼ 0:110 and xir ¼ 0:005 a.u.
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R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
triangles on the plots). Since the potential energy curve for X 1 Rþ state in good approximation is harmonic, one should expect very fast convergence in the ir–vis case. As it is seen from Fig. 2, the convergence is reached after inclusion of two excited vibrational levels. In the vis–ir case there is no summation over the vibrational states of the ground electronic state, so the b values are not sensitive to the vibrational structure in the X 1 Rþ state. ir–vis The dependence of bzzz on infrared frequency ranging from xir ¼ 0:0015 a.u. (329 cm1 ) to xir ¼ 0:0060 a.u. (1317 cm1 ) is presented in Table 2. It can be seen that near the latter limiting value, the bzzz is strongly enhanced. The inspection of Eq. (5) shows that the denominator becomes very small when hxir is close to the energy spacing between the two first vibrational levels of the electronic ground state ( hx0l ). In the case of X 1 Rþ electronic state, the energy gap between m ¼ 1 and m ¼ 0 vibrational levels is 1353 cm1 . Hence, when increasing the infrared angular frequency the (x0l xir ) term becomes small and the value of ir–vis bzzz increases substantially. The dependence of bzzz on the visible frequency is presented in Table 3. Obviously, the increase of the visible frequency leads to an increase of the bzzz values. Since the energy difference between ground vibrational states of X 1 Rþ and A1 R is 0.1169 a.u., the values of xvis were chosen to be far from the electronic resonance and approach it gradually. Analogously to the infrared-visible term, the key
Table 3 bir–vis and bvis–ir values as a function of the visible frequency zzz zzz xvis
bir–vis zzz
bvis–ir zzz
0.080 0.085 0.090 0.092 0.096 0.100 0.105 0.110
)7.4 )8.5 )10.1 )10.9 )12.7 )15.4 )20.3 )29.6
)2009 )2514 )3241 )3626 )4639 )6165 )9507 )17 015
xir ¼ 0:001 a.u. All values are in a.u. vis–ir with the factor responsible for the increase of bzzz frequencies is the decrease of the denominator value. The comparison of the absolute values of both bzzz contributions brings about the problem of explaining the observed large differences, i.e. the bvis–ir values are usually larger than their bir–vis analogs. In order to explain this difference, we partitioned each of the expansions defined by Eqs. (4) and (5) into two components (Table 4). The first one comprises the diagonal contributions denoted as kk (k ¼ l) while the second one comprised the off-diagonal contributions kl (k 6¼ l). In the infrared-visible case, the off-diagonal part of b is significantly larger than the diagonal contribution. On the other hand, a totally different pattern is observed for the visible-infrared term where both kk and kl contributions are of similar magnitude. The diagonal term of bvis–ir reads vis–ir bzzz ¼
Table 2 vis–ir bir–vis and bzzz values as a function of the infrared frequency zzz xir 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060
(329.2) (438.9) (548.7) (658.4) (768.2) (877.9) (987.6) (1097.4) (1207.1) (1316.8)
bir–vis zzz
bvis–ir zzz
)35 )42 )51 )65 )84 )114 )167 )271 )550 )2607
)17 662 )18 366 )19 137 )19 987 )20 930 )21 987 )23 181 )24 553 )26 155 )28 077
xvis ¼ 0:110 a.u. All values are in a.u. except those in parentheses which are given in cm1 .
0 1 X DlEk;00 h2 k
h0; 0j^ lz jE; kihk; Ej^ lz j0; 0i ; ðxkE xvis xir ÞðxkE xvis Þ
ð7Þ
where DlEk;00 stands for the z-component of the dipole moment difference between the kth vibra-
Table 4 Partitioning of the b values into their diagonal (kk) and off-diagonal (kl) terms
bir–vis zzz bvis–ir zzz
kk
kl
Total
)20 )10 816
)251 )13 737
)271 )24 553
xvis ¼ 0:110 and xir ¼ 0.005 a.u. All values are in a.u.
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
tional state of the electronic excited state E and the ground electronic state with m ¼ 0. This diagonal contribution is of the same order of magnitude as
[a.u.]
ωvis = 0.110 a.u. 2.0e+05
vis-ir
8.0e+04
β
β
its off-diagonal counterpart that is proportional to hk; Ej^ lz jE; li, of which the leading harmonic part is proportional to the derivative of the E-excited
ωvis = 0.110 a.u.
ir-vis
[a.u.]
1.2e+05
4.0e+04
0 0.001
0.003
0.006
0.009
1.0e+05
0.001
infrared frequency [a.u.]
[a.u.]
9.0e+05
0.006
0.009
ωvis = 0.115 a.u.
6.0e+05
β
vis-ir
3.0e+07
0.003
infrared frequency [a.u.]
ωvis = 0.115 a.u.
ir-vis
[a.u.]
4.5e+07
β
553
1.5e+07
0 0.001
3.0e+05
0 0.001
0.003 0.006 0.009 infrared frequency [a.u.]
1.2e+07
ωvis = 0.120 a.u.
0.003 0.006 0.009 infrared frequency [a.u.]
ωvis = 0.120 a.u.
[a.u.] β
2.0e+06 0 0.001
8.0e+06
vis-ir
4.0e+06
β
ir-vis
[a.u.]
6.0e+06
0.003
0.006
4.0e+06
0 0.001
0.009
infrared frequency [a.u.]
[a.u.] vis-ir
β
β
4.0e+05
0.003
0.006
0.009
ωvis = 0.125 a.u.
8.0e+05
0 0.001
0.006
infrared frequency [a.u.]
ωvis = 0.125 a.u.
ir-vis
[a.u.]
1.2e+06
0.003
0.009
8.0e+06
4.0e+06
0 0.001
0.003
0.006
Fig. 3. Dispersion of bir–vis and bvis–ir as a function of xir for different values of xvis . zzz zzz
0.009
554
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
state dipole moment with respect to the stretching vibrational normal coordinate. On the other hand, hl; 0j^ lz j0; 0i is the component of the diagonal and off-diagonal bir–vis numerators which differ with respect to the bvis–ir numerator. Its value, which is about two orders of magnitude smaller than hk; Ej^ lz jE; li, accounts largely for the difference of magnitude between bir–vis and bvis–ir . The remaining factors which influence the relative magnitudes of bir–vis and bvis–ir are related to overlaps between vibrational wavefunctions as well as to mechanical and electrical anharmonicities. In order to evaluate b under resonance it was compulsory to include in the treatment a damping factor (C). A value of 10 cm1 has been chosen for all vibronic states. In Fig. 3 we present the real part of the two dominant components of DR-SFG b: Re bir–vis 1 X0X0 ¼ 2 h k l " h0; 0j^ lz jE; kihk; Ejl^z j0; lihl; 0j^ lz j0; 0i 2
2
½ðxkE xr Þ þ 14 C2kE ½ðxl0 xir Þ þ 14 C2l0 # 1 ½ðxkE xr Þðxl0 xir Þ CkE Cl0 ð8Þ 4 Re bvis–ir 1 X0X0 h2 k l " h0; 0j^ lz jE; kihk; Ejl^z jE; lihE; lj^ lz j0; 0i
spectra as well as for different values of xvis . On the other hand, for the vis–ir process, the resonances with the successive vibrational levels of the excited electronic state are clearly visible. For both processes, changing the xvis frequency leads to variations of b that can attain several orders of magnitude.
4. Summary and outlook DR-SFG second-order polarizabilities have been reported for the lithium hydride molecule. The calculations refer to the diagonal longitudinal component of b, ignore rotation, and are based on data calculated using the multireference configuration interaction method with singles and doubles. The convergence with respect to the number of vibrational states as well as the dependence upon both infrared and visible frequencies have been addressed. The total DR-SFG response has been partitioned into its infrared–visible and visible–infrared dominant components and it has been shown that the later is the largest. This has been related to the similar magnitude of the diagonal (kk) and off-diagonal (kl) contributions to the visible–infrared term. Extension to larger systems, including conjugated organic compounds, of such ab initio treatments to DR-SFG is expected to be a useful tool to interpret the physics behind complex DR-SFG spectra [4].
¼
2
½ðxkE xr Þ þ
1 2 C ½ðxlE 4 kE
2
xvis Þ þ
Acknowledgements
1 2 C 4 lE
#
1 ½ðxkE xr ÞðxlE xvis Þ CkE ClE : 4
ð9Þ
For xvis values ranging from 0.110 to 0.125 a.u., the bir–vis spectra display a resonance at 0.0062 a.u. which corresponds to the m ¼ 0 m ¼ 1 vibrational transition of the ground electronic state. In addition, there are bands separated by approximately 0.0015 a.u. that correspond to resonances with successive vibrational levels of the excited electronic state. However, the intensities of these transitions can differ strongly within the same
The authors thank Professor Andrzej J. Sadlej for his help and advices. One of the authors (RZ) gratefully acknowledges the financial support from CGRI (Commissariat General aux Relations Internationales) during his stay in Namur. BC thanks the Belgian National Fund for Scientific Research for his Senior Research Associate position. We acknowledge the support from the Interuniversity Attraction Poles Programme on ÔSupramolecular Chemistry and Supramolecular Catalysis (IUAP no. P5-03)Õ–Belgian State-Federal Office for Scientific, Technical and Cultural Affairs. This work was facilitated in part by Wroclaw
R. Zalesny et al. / Chemical Physics Letters 380 (2003) 549–555
University of Technology. We also acknowledge the use of the Sun Enterprise 6000 at the University Center for Information Technology of the Nicolaus Copernicus University. Part of the calculations have also been performed on the Pentium III/Pentium IV cluster of the CTA lab for which the authors acknowledge the financial support of the FNRS. References [1] Z. Chen, Y.R. Shen, G.A. Somorjai, Annu. Rev. Phys. Chem. 53 (2002) 437. [2] S. Baldelli, N. Markovic, P. Ross, Y.R. Shen, G.A. Somorjai, J. Phys. Chem. B 103 (1999) 8920. [3] X. Su, P.S. Cremer, Y.R. Shen, G.A. Somorjai, J. Am. Chem. Soc. 119 (1997) 3994. [4] C. Humbert, L. Dreesen, S. Nihonyanagi, T. Masuda, T. Kondo, A.A. Mani, K. Uosaki, P.A. Thiry, A. Peremans, Appl. Surf. Sci. 212–213 (2003) 797. [5] J.Y. Huang, Y.R. Shen, Phys. Rev. A 49 (1994) 3973. [6] S.H. Lin, M. Hayashi, R. Islampour, J. Yu, D.Y. Yang, G.Y.C. Wu, Physica B 222 (1996) 191. [7] M. Hayashi, S.H. Lin, M.B. Raschke, Y.R. Shen, J. Phys. Chem. A 106 (2002) 2271. [8] K.K. Docken, J. Hinze, J. Chem. Phys. 57 (1972) 4928.
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