- Email: [email protected]
Molecular localized excitation by optimal fields
Spectrochimica Acta Part A 55 (1999) 2399 – 2402 www.elsevier.nl/locate/saa
Molecular localized excitation by optimal fields Chen Yang, Ji-xin Cheng *, Qing-shi Zhu Laboratory of Bond-selecti6e Chemistry, Department of Chemical Physics, Uni6ersity of Science and Technology of China, Hefei, Anhui, 230026, China Received 15 September 1998; received in revised form 27 January 1999; accepted 27 January 1999
Abstract With the optimal control theory, a general model is built up for selective excitation of a vibrational wave packet in the eigenstate representation. The control equation in terms of Hilbert-space wave function evolution is obtained and the form of the globally optimal field in the weak response regime is derived. Based on the intramolecular dynamics information obtained from high-resolution spectroscopy, numerical implementations for selective excitation of the C–H bond in the CF3H molecule are carried out. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Optimal control; Wave packet; CF3H; Local mode; Bond-selective
A prerequisite of realizing bond-selective chemistry is to prepare a state that localizes energy in some part of the reagent molecule [1]. Selective breaking of chemical bonds has been achieved by localized vibrational excitation in a specific bond of HOD [2,3], or by localized electronic excitation in CH2IBr and CH3SH [4 – 6]. This method is, however, called passive control because it does not make use of the coherent property of light. In the last decade various scenarios for active control of molecular dynamics have been developed. Brumer and Shapiro proposed a coherent control theory based on the principle of quantum mechanical interference between competing reaction paths [7]. Tannor and Rice suggested a pumpdump control scheme that utilizes a sequence of * Corresponding author. Tel.: + 86-551-3603748; fax: +86551-3602969.
light pulses to drive a molecule to a desired state [8,9]. The general optimal control theory was employed by Rabitz [10], Kosloff [11], Yan [12] and their co-workers in order to search for the optimal laser fields that maximize the control yield at a specific time under certain constraints. On the other hand, the high resolution and high sensitivity spectroscopy has provided rich information of isolated intramolecular dynamics in the high vibrational energy region [13,14]. The optimal control theory together with the spectroscopic information enables us to find suitable laser fields that excite a specific bond. The aim of control in the present work is to produce a wave packet corresponding to single bond vibration (or so-called local mode vibration). We shall first present the general form of the optimal field for controlling a vibrational wave packet and then implement it in the selective
1386-1425/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 1 4 2 5 ( 9 9 ) 0 0 0 2 9 - 3
&
C. Yang et al. / Spectrochimica Acta Part A 55 (1999) 2399–2402
2400
excitation of the C – H bond in the CF3H molecule. Let us consider a molecular vibrational system which consists of the ground state g and n excited states ei , i= 1, 2…n. The molecule is initially in the ground state with energy 'vg. The field-molecule Hamiltonian in the presence of an external field o(t) can be written as
tf
G(tf, t)d H(t)G(t, t0) dt,
t0
we have
dA(tf)= 2 Re (− i/')
(7)
&
tf
dtc(tf) f
t0
n
f( t tf) d H(t) c(t)
(1)
H(t)= Hm −Do(t) Here, Hm is the vibrational Hamiltonian, n
Hm ='vg gg + % '(vi +vg) ei ei ,
(2)
i=1
where vi is the transition frequency from g to ei . The transition dipole operator assumes the following form, n
D = % mi ei g +h.c.
(3)
i=1
Our goal is to excite a desired wave packet, f =ni= 1 ai ei , at a specific time tf. The target operator can be written as A. = ff . In order to find the optimal excitation field, we consider the control functional [12], 1 J(tf)=A(tf) − l 2
dG(tf, t0)= (− i/')
&
tf
dt o 2(t)
(4)
t0
A(tf) is the expectation value of the target operator at time tf, A(tf)=c(tf) ff c(tf)
d J(tf)= d A(tf) −l
&
tf
dt o(t)d o(t)
t0
By exploiting variation of Green functions,
According to the total Hamiltonian in Eq. (1), we have dH(t)= − Ddo(t). c(t) and f(t, tf) is the forward- and backward-propagated wave functions for the system and target, respectively. By substituting Eq. (8) into Eq. (6) and applying the variation principle, dJ(tf)= 0, we obtain the control equation for the optimal field as [16] 2 Re[c*(tf)f(t; tf)]= lo(t)
(6)
(9)
Note that we have introduced the complex control yield c(tf)= f c(tf ), and the Hilbert-space control kernel, f(t; tf)= i/'f(t, tf) D c(t). In the weak response regime, we only consider the zero-order control kernel, f (0)(t; tf)= (i/')f (0)(t, tf) D c (0)(t) = (i/') e − ivg(tf − t0)fe(tf − t),
(10)
and the first-order control yield, c (1)(tf)= f c (1)(tf) = (i/') e − ivg(tf − t0)
(5)
Here, c(tf) =G(tf, t0) c(t0), where the Hilbertspace propagator G is governed by the total Hamiltonian H(t). The second term in the right side of Eq. (4) is the penalty function characterized by a constant weight l. In our case, the penalty is simply the constraint on the field energy. If a certain shape of control field is favored in the optimization process the constant l should be replaced by a time-dependent cost penalty function l(t) [10,12,15]. The variation of J(tf) with respect to o(t) is given by
(8)
&
tf
fe(tf − t)o(t) dt
(11)
t0
fe(t) in Eqs. (10) and (11) is given by fe(t)= f e − i(Hm − 'vg)t/'D g
(12) (1)
It can be proved that the phase of c (tf) has no effect on the control dynamics and can be set as zero [16]. Combining Eqs. (9)–(11), we obtain the simple form of the globally optimal field in the weak response regime [8,17], o(t)8 Re[f e − i(Hm − 'vg)(tf − t)/'D g]
(13)
Upon substitution of f and D with their expressions, we arrive at the final form of the optimal field for exciting the wave packet f, n
o(t)8 % aimi [e − ivi (tf − t) + eivi (tf − t)] i=1
(14)
C. Yang et al. / Spectrochimica Acta Part A 55 (1999) 2399–2402
Eq. (14) indicates that o(t) can be viewed as coherent superposition of n cw components. Each component excites a particular eigenstate ei and has defined intensity of (aimi )2 and phase of vitf. The field o(t) in Eq. (14) drives the molecular system optimally to the target wave packet f at time tf. Noted that Eq. (14) is similar to the previous results of Rabitz et al. [18,19], who derived the weak-response form of the optimal field for a d-target in the coordinate space. The CF3H molecule is a good example of rapid intramolecular vibrational relaxation via Fermi resonance between the CH stretching mode s and a bending mode b [20]. In a Hamiltonian block with the chromophore quantum number N =6s + 6b /2, we have N + 1 zero-order states, 6s, 6b, lb = 0, 6s = 0, 1,… N, that are connected by Fermi coupling. By diagonalizing a triangle Hamiltonian, one can obtain N + 1 eigenstates, Nm , m= 0, 1…N. For the N =3 polyad, according to Du¨bal and Quack’s experiments and calculations [20], the C– H stretching state 3, 0, 0 can be written as
2401
Fig. 1. The globally optimal field for selective excitation of the C – H bond in the CF3H molecule.
3, 0, 0 = 0.727 30 −0.657 31 +0.199 32 −0.030 33
(15)
The transition frequencies and the relative intensities (gi 8 m 2i ) of the 3m (m = 0, 1, 2 and 3) bands are listed in Table 1. With the information in Eq. (15) and Table 1, we can compute the optimal field for selective excitation of the C – H bond in CF3H by Eq. (14). For the case of tf =100 fs, the form of the optimal field is shown in Fig. 1. In the presence of the external field, the population P in the wave packet f= 3, 0, 0 is determined by P = f e − iHt/' g 2, here H is the time-dependent Hamiltonian in Eq. (1). The control quality is evaluated by the achievement function defined as [12] Table 1 The band centers (vi ) and relative intensities (gexp 8m 2i ) for i the N= 3 polyad of the CF3H molecule j vj cm−1 gexp j
1 8792.70 0.52
2 8589.28 0.45
3 8286.0 0.03
4 7890 B0.01
Fig. 2. The achievement a(t) versus time for the globally optimal field.
a 2(t)= f ce(t) 2/[f fce(tf) ce(tf)]
(16)
Here, ce(t) is the wave function in the excited states. Fig. 2 shows the achievement a(t) for the optimal field. It can be seen that a increases gradually with time and is close to 1.0 at the target time. This indicates that almost all the vibrational energy is localized in the C–H bond.
Acknowledgements The support of the National Natural Science Foundation of China is gratefully acknowledged. The authors thank Professor Yijing Yan in Hong
2402
C. Yang et al. / Spectrochimica Acta Part A 55 (1999) 2399–2402
Kong University of Science and Technology for helpful discussions and a critical reading of the manuscript.
References [1] R.N. Zare, Science 279 (1998) 875. [2] A. Sinha, R.L. Vander Wal, F.F. Crim, J. Chem. Phys. 92 (1990) 401. [3] A. Sinha, J.D. Thoemke, F.F. Crim, J. Chem. Phys. 96 (1992) 372. [4] L.J. Butler, E.J. Hintsa, Y.T. Lee, J. Chem. Phys. 84 (1986) 4104. [5] L.J. Butler, E.J. Hintsa, S.F. Shane, Y.T. Lee, J. Chem. Phys. 86 (1987) 2051. [6] J.S. Keller, P.W. Kash, E. Jensen, L.J. Butler, J. Chem. Phys. 96 (1992) 4324.
.
[7] M. Shapiro, P. Brumer, Int. Rev. Phys. Chem. 13 (1994) 187. [8] D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 (1985) 5013. [9] D.J. Tannor, R. Kosloff, S.A. Rice, J. Chem. Phys. 85 (1986) 5805. [10] H. Rabitz, S. Shi, Adv. Mol. Vib. Collis. Dyn. 1A (1991) 187. [11] R. Kosloff, S.A. Rice, P. Gaspard, S. Tersigni, D.J. Tannor, Chem. Phys. 139 (1989) 201. [12] Y.J. Yan, R.E. Gillilan, R.M. Whitnell, K.R. Wilson, S. Mukamel, J. Phys. Chem. 97 (1993) 2320. [13] M. Quack, Annu. Rev. Phys. Chem. 41 (1990) 493. [14] K.K. Lehmann, G. Scoles, B.H. Pate, Annu. Rev. Chem. Phys. 45 (1994) 241. [15] S. Shi, H. Rabitz, Chem. Phys. 139 (1989) 185. [16] Y.J. Yan, Z.W. Shen, Y. Zhao, Chem. Phys. 233 (1998) 191. [17] Y.J. Yan, Annu. Rep. (R. Soc. Chem.) C94 (1998) 397. [18] V. Dubov, H. Rabitz, Chem. Phys. Lett. 235 (1995) 309. [19] V. Dubov, H. Rabitz, J. Chem. Phys. 103 (1995) 8412. [20] H.R. Dubal, M. Quack, J. Chem. Phys. 81 (1984) 3779.
Molecular localized excitation by optimal fields Chen Yang, Ji-xin Cheng *, Qing-shi Zhu Laboratory of Bond-selecti6e Chemistry, Department of Chemical Physics, Uni6ersity of Science and Technology of China, Hefei, Anhui, 230026, China Received 15 September 1998; received in revised form 27 January 1999; accepted 27 January 1999
Abstract With the optimal control theory, a general model is built up for selective excitation of a vibrational wave packet in the eigenstate representation. The control equation in terms of Hilbert-space wave function evolution is obtained and the form of the globally optimal field in the weak response regime is derived. Based on the intramolecular dynamics information obtained from high-resolution spectroscopy, numerical implementations for selective excitation of the C–H bond in the CF3H molecule are carried out. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Optimal control; Wave packet; CF3H; Local mode; Bond-selective
A prerequisite of realizing bond-selective chemistry is to prepare a state that localizes energy in some part of the reagent molecule [1]. Selective breaking of chemical bonds has been achieved by localized vibrational excitation in a specific bond of HOD [2,3], or by localized electronic excitation in CH2IBr and CH3SH [4 – 6]. This method is, however, called passive control because it does not make use of the coherent property of light. In the last decade various scenarios for active control of molecular dynamics have been developed. Brumer and Shapiro proposed a coherent control theory based on the principle of quantum mechanical interference between competing reaction paths [7]. Tannor and Rice suggested a pumpdump control scheme that utilizes a sequence of * Corresponding author. Tel.: + 86-551-3603748; fax: +86551-3602969.
light pulses to drive a molecule to a desired state [8,9]. The general optimal control theory was employed by Rabitz [10], Kosloff [11], Yan [12] and their co-workers in order to search for the optimal laser fields that maximize the control yield at a specific time under certain constraints. On the other hand, the high resolution and high sensitivity spectroscopy has provided rich information of isolated intramolecular dynamics in the high vibrational energy region [13,14]. The optimal control theory together with the spectroscopic information enables us to find suitable laser fields that excite a specific bond. The aim of control in the present work is to produce a wave packet corresponding to single bond vibration (or so-called local mode vibration). We shall first present the general form of the optimal field for controlling a vibrational wave packet and then implement it in the selective
1386-1425/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 1 4 2 5 ( 9 9 ) 0 0 0 2 9 - 3
&
C. Yang et al. / Spectrochimica Acta Part A 55 (1999) 2399–2402
2400
excitation of the C – H bond in the CF3H molecule. Let us consider a molecular vibrational system which consists of the ground state g and n excited states ei , i= 1, 2…n. The molecule is initially in the ground state with energy 'vg. The field-molecule Hamiltonian in the presence of an external field o(t) can be written as
tf
G(tf, t)d H(t)G(t, t0) dt,
t0
we have
dA(tf)= 2 Re (− i/')
(7)
&
tf
dtc(tf) f
t0
n
f( t tf) d H(t) c(t)
(1)
H(t)= Hm −Do(t) Here, Hm is the vibrational Hamiltonian, n
Hm ='vg gg + % '(vi +vg) ei ei ,
(2)
i=1
where vi is the transition frequency from g to ei . The transition dipole operator assumes the following form, n
D = % mi ei g +h.c.
(3)
i=1
Our goal is to excite a desired wave packet, f =ni= 1 ai ei , at a specific time tf. The target operator can be written as A. = ff . In order to find the optimal excitation field, we consider the control functional [12], 1 J(tf)=A(tf) − l 2
dG(tf, t0)= (− i/')
&
tf
dt o 2(t)
(4)
t0
A(tf) is the expectation value of the target operator at time tf, A(tf)=c(tf) ff c(tf)
d J(tf)= d A(tf) −l
&
tf
dt o(t)d o(t)
t0
By exploiting variation of Green functions,
According to the total Hamiltonian in Eq. (1), we have dH(t)= − Ddo(t). c(t) and f(t, tf) is the forward- and backward-propagated wave functions for the system and target, respectively. By substituting Eq. (8) into Eq. (6) and applying the variation principle, dJ(tf)= 0, we obtain the control equation for the optimal field as [16] 2 Re[c*(tf)f(t; tf)]= lo(t)
(6)
(9)
Note that we have introduced the complex control yield c(tf)= f c(tf ), and the Hilbert-space control kernel, f(t; tf)= i/'f(t, tf) D c(t). In the weak response regime, we only consider the zero-order control kernel, f (0)(t; tf)= (i/')f (0)(t, tf) D c (0)(t) = (i/') e − ivg(tf − t0)fe(tf − t),
(10)
and the first-order control yield, c (1)(tf)= f c (1)(tf) = (i/') e − ivg(tf − t0)
(5)
Here, c(tf) =G(tf, t0) c(t0), where the Hilbertspace propagator G is governed by the total Hamiltonian H(t). The second term in the right side of Eq. (4) is the penalty function characterized by a constant weight l. In our case, the penalty is simply the constraint on the field energy. If a certain shape of control field is favored in the optimization process the constant l should be replaced by a time-dependent cost penalty function l(t) [10,12,15]. The variation of J(tf) with respect to o(t) is given by
(8)
&
tf
fe(tf − t)o(t) dt
(11)
t0
fe(t) in Eqs. (10) and (11) is given by fe(t)= f e − i(Hm − 'vg)t/'D g
(12) (1)
It can be proved that the phase of c (tf) has no effect on the control dynamics and can be set as zero [16]. Combining Eqs. (9)–(11), we obtain the simple form of the globally optimal field in the weak response regime [8,17], o(t)8 Re[f e − i(Hm − 'vg)(tf − t)/'D g]
(13)
Upon substitution of f and D with their expressions, we arrive at the final form of the optimal field for exciting the wave packet f, n
o(t)8 % aimi [e − ivi (tf − t) + eivi (tf − t)] i=1
(14)
C. Yang et al. / Spectrochimica Acta Part A 55 (1999) 2399–2402
Eq. (14) indicates that o(t) can be viewed as coherent superposition of n cw components. Each component excites a particular eigenstate ei and has defined intensity of (aimi )2 and phase of vitf. The field o(t) in Eq. (14) drives the molecular system optimally to the target wave packet f at time tf. Noted that Eq. (14) is similar to the previous results of Rabitz et al. [18,19], who derived the weak-response form of the optimal field for a d-target in the coordinate space. The CF3H molecule is a good example of rapid intramolecular vibrational relaxation via Fermi resonance between the CH stretching mode s and a bending mode b [20]. In a Hamiltonian block with the chromophore quantum number N =6s + 6b /2, we have N + 1 zero-order states, 6s, 6b, lb = 0, 6s = 0, 1,… N, that are connected by Fermi coupling. By diagonalizing a triangle Hamiltonian, one can obtain N + 1 eigenstates, Nm , m= 0, 1…N. For the N =3 polyad, according to Du¨bal and Quack’s experiments and calculations [20], the C– H stretching state 3, 0, 0 can be written as
2401
Fig. 1. The globally optimal field for selective excitation of the C – H bond in the CF3H molecule.
3, 0, 0 = 0.727 30 −0.657 31 +0.199 32 −0.030 33
(15)
The transition frequencies and the relative intensities (gi 8 m 2i ) of the 3m (m = 0, 1, 2 and 3) bands are listed in Table 1. With the information in Eq. (15) and Table 1, we can compute the optimal field for selective excitation of the C – H bond in CF3H by Eq. (14). For the case of tf =100 fs, the form of the optimal field is shown in Fig. 1. In the presence of the external field, the population P in the wave packet f= 3, 0, 0 is determined by P = f e − iHt/' g 2, here H is the time-dependent Hamiltonian in Eq. (1). The control quality is evaluated by the achievement function defined as [12] Table 1 The band centers (vi ) and relative intensities (gexp 8m 2i ) for i the N= 3 polyad of the CF3H molecule j vj cm−1 gexp j
1 8792.70 0.52
2 8589.28 0.45
3 8286.0 0.03
4 7890 B0.01
Fig. 2. The achievement a(t) versus time for the globally optimal field.
a 2(t)= f ce(t) 2/[f fce(tf) ce(tf)]
(16)
Here, ce(t) is the wave function in the excited states. Fig. 2 shows the achievement a(t) for the optimal field. It can be seen that a increases gradually with time and is close to 1.0 at the target time. This indicates that almost all the vibrational energy is localized in the C–H bond.
Acknowledgements The support of the National Natural Science Foundation of China is gratefully acknowledged. The authors thank Professor Yijing Yan in Hong
2402
C. Yang et al. / Spectrochimica Acta Part A 55 (1999) 2399–2402
Kong University of Science and Technology for helpful discussions and a critical reading of the manuscript.
References [1] R.N. Zare, Science 279 (1998) 875. [2] A. Sinha, R.L. Vander Wal, F.F. Crim, J. Chem. Phys. 92 (1990) 401. [3] A. Sinha, J.D. Thoemke, F.F. Crim, J. Chem. Phys. 96 (1992) 372. [4] L.J. Butler, E.J. Hintsa, Y.T. Lee, J. Chem. Phys. 84 (1986) 4104. [5] L.J. Butler, E.J. Hintsa, S.F. Shane, Y.T. Lee, J. Chem. Phys. 86 (1987) 2051. [6] J.S. Keller, P.W. Kash, E. Jensen, L.J. Butler, J. Chem. Phys. 96 (1992) 4324.
.
[7] M. Shapiro, P. Brumer, Int. Rev. Phys. Chem. 13 (1994) 187. [8] D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 (1985) 5013. [9] D.J. Tannor, R. Kosloff, S.A. Rice, J. Chem. Phys. 85 (1986) 5805. [10] H. Rabitz, S. Shi, Adv. Mol. Vib. Collis. Dyn. 1A (1991) 187. [11] R. Kosloff, S.A. Rice, P. Gaspard, S. Tersigni, D.J. Tannor, Chem. Phys. 139 (1989) 201. [12] Y.J. Yan, R.E. Gillilan, R.M. Whitnell, K.R. Wilson, S. Mukamel, J. Phys. Chem. 97 (1993) 2320. [13] M. Quack, Annu. Rev. Phys. Chem. 41 (1990) 493. [14] K.K. Lehmann, G. Scoles, B.H. Pate, Annu. Rev. Chem. Phys. 45 (1994) 241. [15] S. Shi, H. Rabitz, Chem. Phys. 139 (1989) 185. [16] Y.J. Yan, Z.W. Shen, Y. Zhao, Chem. Phys. 233 (1998) 191. [17] Y.J. Yan, Annu. Rep. (R. Soc. Chem.) C94 (1998) 397. [18] V. Dubov, H. Rabitz, Chem. Phys. Lett. 235 (1995) 309. [19] V. Dubov, H. Rabitz, J. Chem. Phys. 103 (1995) 8412. [20] H.R. Dubal, M. Quack, J. Chem. Phys. 81 (1984) 3779.