- Email: [email protected]
Interaction energy of coupled permanent dipoles in oriented molecules
Chemical Physics Letters 496 (2010) 362–364
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Interaction energy of coupled permanent dipoles in oriented molecules Michele Battezzati a, Valerio Magnasco b,* a b
Via Bernardino Drovetti 7, 10138 Torino, Italy Dipartimento di Chimica e Chimica Industriale dell’Università, Via Dodecaneso 31, 16146 Genova, Italy
a r t i c l e
i n f o
Article history: Received 21 June 2010 In final form 16 July 2010 Available online 21 July 2010
a b s t r a c t The partition function of coupled molecular dipoles in a static electric field parallel to the supermolecular axis is evaluated by expanding the interaction energy function around the equilibrium position of minimum potential energy. The resulting expansion of the configurational integral is, as was proved by these same authors, an asymptotic representation in the limit of the strong interaction parameter a. This allows for the computation of one-sided averages of the supermolecular dipole moment not attainable through symmetrical procedures. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Long time ago Keesom [1] proposed a simple formula yielding the interaction energy of coupled molecular dipoles as a function of distance R and absolute temperature T, which results proportional to the sixth inverse power of the distance. At higher dipole strengths and lower temperatures, or shorter separation, Keesom’s calculations loose their validity because of more pronounced mutual orientational effects, owing to the more effective correlation between dipole orientation. As a consequence of this, we evaluated the interaction energy under the assumption of dipoles oscillating around only one of the two equivalent stable equilibrium positions, which are separated by a barrier [2–5], denoting by E that one of these regions which contains the origin of the expansion. Since the interaction potential energy between the dipoles was expanded around one of these equilibrium points, the configurational probability density cannot be represented adequately far from this point with a finite number of terms, for jaj ? 1, where a is the interaction parameter (see Eq. (20 )), because convergence is not uniform with respect to a in the semi-infinite interval 0 6 jaj < +1. However, asymptotic convergence of the series expansion of the integral was proved in [3], while in [4] it was proved that the configurational integral, extended to the region external to E, of a finite number of terms of the series, vanishes asymptotically in the limit a ? 1. Consequently, the integral extended to the whole space of the variable coordinates of the series expansion truncated after a finite number of terms converges to the true integral over E without altering its limiting value. This situation is substantially unchanged by the introduction of a uniform electric field, which orients the supermolecule preferably in the direction of the applied field, in order to minimize the electrostatic energy. Aim of this Let* Corresponding author. Fax: +39 010 3536199. E-mail address: [email protected] (V. Magnasco). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.07.064
ter is the evaluation of the average energy of the supermolecule in an electric field of strength e, so allowing for the computation of one-sided averages which are not attainable through symmetrical procedures. 2. Keesom integral in electrostatic field We proceed to the evaluation of Keesom integral, or configurational free energy, in a uniform electric field, assuming thermal equilibrium, for a supermolecule formed by two identical dipolar molecules A and B. It is introduced the potential energy functions V (X) and W (X), where X (#A,#B,/) is the set of three internal coordinates defining the mutual orientation of dipoles. It is assumed that the supermolecular axis R is parallel to the external field e, therefore
VðXÞ ¼ bkTðcos #A þ cos #B Þ b¼
le
ð10 Þ
kT
WðXÞ ¼ akTðsin #A sin #B cos / 2 cos #A cos #B Þ a¼
ð1Þ
l2
ð2Þ ð20 Þ
R3 kT
where / is the angle between the planes containing R and the dipoles lA, lB, respectively, and #A,#B are the polar angles of each dipole with respect to R. Therefore it is defined the integral over configuration space
Kða; bÞ ¼
Z
VðXÞ þ WðXÞ dX exp kT
dX ¼ sin #A sin #B d#A d#B d/
ð3Þ ð30 Þ
363
M. Battezzati, V. Magnasco / Chemical Physics Letters 496 (2010) 362–364
Lastly, we obtain the one-sided average in the metastable state as
3. Evaluation of K (a, b) The asymptotic evaluation of the Keesom integral (3) in the limit of large values of jaj was already accomplished in Refs. [2,4,5], a different approach being proposed in [6]. Our procedure consists in introducing vector planar coordinates #A, #B and in writing [4]:
d/A dX ¼ d#A d#B
sin #A sin #B #A #B
ð4Þ
where /A is a dummy variable denoting the rotational symmetry of the supermolecule around R as a whole, which contributes a factor 2p. Therefore we define the one-sided Keesom integral in the metastable state as
Kða; bÞ ¼
1 2p
ZZ
sin #A sin #B #A #B j#A jþj#B j6p VðXÞ þ WðXÞ exp kT d#A d#B
ð5Þ
We now expand the trigonometric functions in powers of #A ; #B , upon defining the normal coordinates a, b:
a ¼ #A þ #B
ð6Þ
b ¼ #A þ #B the Jacobian of the transformation being
@ða; bÞ ¼4 @ð#A ; #B Þ
ð60 Þ
ZZ
1 2 dadbqða; bÞ 1 ða þ b2 Þ þ h:o:t: 12 j#A jþj#B j6p ð7Þ
ð12Þ
which gives the energy correction at small values of the field strength. If the alignment is not complete it should be valuable to evaluate the average energy of the supermolecule in an electric field pointing in a direction oblique with respect to the axis R. If the deviation angle c is not too large, taking the plane containing R and the electric field e as reference plane containing the axis g, from which rotations /A and /B are measured, the Keesom integral is written
1 dadb exp 2ða þ bÞ þ ð3a þ bÞa2 4 all space 1 2p 2 þ ða þ bÞb þ cag 4 ð3a þ bÞða þ bÞ c2 exp 2ða þ bÞ þ h:o:t: 3a þ b
c¼
1 8p
ZZ
eg l
ð13Þ
ð130 Þ
kT
2
Kða; b; cÞ
ð8Þ
2pk T 2 R6
l2 ð3l þ eR3 Þðl þ eR3 Þ " 2 l2 el þ 3 exp kT
R
3l2 ec2
#
kTð3l þ eR3 Þ
ð14Þ
From Eq. (13) follows the average interaction energy with the oblique electric field:
4. Results Eq. (7) can be evaluated to the desired accuracy, yielding an expansion in inverse powers of 3a + b and a + b. It was proved in Ref. [4] that any sum of a finite number of terms of this expansion pffiffiffi converges exponentially to zero outside of a circle of radius p= 2. Furthermore, it was proved in [3] that the integral extended to the region satisfying j#Aj + j #Bj 6 p converges asymptotically to the Keesom integral inside of that region, which is therefore the limit of the integral extended to the whole space:
Kða; bÞ
2 hVðXÞi 2le 1 þ 3a
At small values of deviation angle c Eq. (13) yields, using Eqs. (10 ), (20 ), and (130 )
where q(a, b) is obtained by expansion of Eq. (3):
1 1 qða; bÞ ¼ exp 2ða þ bÞ þ ð3a þ bÞa2 þ ða þ bÞb2 4 4 12a þ b 4 4a þ b 4 b 2 2 b a ab 1 192 192 32 4a þ b j a bj2 þ h:o:t: 48
ð11Þ
Putting b = 0 into the bracket, we obtain:
Kða; b; cÞ
Then Eq. (5) is recast into the form
1 Kða; bÞ ¼ 8p
@ hVðXÞi ¼ bkT ln Kða; bÞ @b " 2 1 1 5a2 þ 4ab þ b þ þ le 2 þ 2 3a þ b a þ b 3ð3a þ bÞ ða þ bÞ2 # 2 2 63a2 þ 24ab þ b 7a2 þ 8ab þ b þ h:o:t: 6ð3a þ bÞ4 6ða þ bÞ4
2p exp½2ða þ bÞ ð3a þ bÞða þ bÞ " # 2a þ b 12a þ b 4a þ b 1þ þ h:o:t: 3ð3a þ bÞða þ bÞ 6ð3a þ bÞ2 6ða þ bÞ2
@ @ ln Kða; b; cÞ hVðX; /A Þi kT b þ c @b @c " # 2 2 c 1 1 2kTc kTb 2 þ þ þ þ h:o:t: 3a þ b 3a þ b a þ b 3a þ b
ð15Þ
In the limit of small correlation between dipole orientation (or a ? 0) Eq. (15) yields
" hVðX; /A Þi 2le1
1 1þ 2
2
2l j e j þ2kT
eg e1
kT
le1
# þ h:o:t: ð16Þ
Therefore, dipoles which are uncoupled orient preferably toward the field direction, being dis-aligned by thermal fluctuations.
ð9Þ As b ? 0, it may be ascertained that the rhs of Eq. (9) approaches the limiting value which we obtained in Refs. [2,4,5], while for a ? 0 it yields the correct asymptotic value
Kð0; bÞ
2p 2
b
expð2bÞ
ð10Þ
5. Conclusions Eq. (11) allows for the evaluation of the average energy of the supermolecule in the electric field of strength e, showing that the present calculation is suitable for computation of one-sided averages which are not attainable through a symmetrical procedure.
364
M. Battezzati, V. Magnasco / Chemical Physics Letters 496 (2010) 362–364
There has been done recently considerable effort in order to obtain adiabatically oriented molecules in thermal equilibrium [7] in a static electric field, either by ‘brute force’ orienting [8], or by femtosecond laser pulse and preparation of state-selected molecules using a hexapole [9]. Though the efficiency of the alignment is high, nonetheless it may be not complete, therefore calculations have been made including this case. It is pointed out, however, that our calculations refer to idealized experimental conditions, which may differ from the concrete situations resulting from laboratory practice. It is possible in this way to take account also of incomplete alignment and orientation, which appear to be produced in actual experiments, by evaluation of averages over the angular distribution of supermolecular axis around the field direction. Integration of the rhs of Eq. (14) over dc with the proper angular factor gives the configurational free energy taking into account all orientations of the supermolecular axis with respect to the field. Notice that the angular distribution (14) becomes increasingly flat with distance R,
so that dipoles which are far apart can occupy with equal probability all relative positions.
Acknowledgment Support by the University of Genoa is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
W.H. Keesom, Z. Phys. 22 (1921) 129. M. Battezzati, V. Magnasco, J. Phys. A: Math. Gen. 37 (2004) 9677. M. Battezzati, V. Magnasco, J. Phys. A: Math. Gen. 38 (2005) 6715. M. Battezzati, V. Magnasco, C R Chim. 12 (2009) 854. M. Battezzati, V. Magnasco, Chem. Phys. Lett. 492 (2010) 179. P.C. Abbott, J. Phys. A: Math. Theor. 40 (2007) 8599. H. Stapelfeldt, T. Seideman, Rev. Mod. Phys. 75 (2003) 543. H.J. Loesch, A. Remscheid, J. Chem. Phys. 93 (1990) 4779. O. Ghafur, A. Rouzée, A. Gijsbertsen, W. Kiu Siu, S. Stolte, M.J.J. Vrakking, Nat. Phys. 5 (2009) 289.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Interaction energy of coupled permanent dipoles in oriented molecules Michele Battezzati a, Valerio Magnasco b,* a b
Via Bernardino Drovetti 7, 10138 Torino, Italy Dipartimento di Chimica e Chimica Industriale dell’Università, Via Dodecaneso 31, 16146 Genova, Italy
a r t i c l e
i n f o
Article history: Received 21 June 2010 In final form 16 July 2010 Available online 21 July 2010
a b s t r a c t The partition function of coupled molecular dipoles in a static electric field parallel to the supermolecular axis is evaluated by expanding the interaction energy function around the equilibrium position of minimum potential energy. The resulting expansion of the configurational integral is, as was proved by these same authors, an asymptotic representation in the limit of the strong interaction parameter a. This allows for the computation of one-sided averages of the supermolecular dipole moment not attainable through symmetrical procedures. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Long time ago Keesom [1] proposed a simple formula yielding the interaction energy of coupled molecular dipoles as a function of distance R and absolute temperature T, which results proportional to the sixth inverse power of the distance. At higher dipole strengths and lower temperatures, or shorter separation, Keesom’s calculations loose their validity because of more pronounced mutual orientational effects, owing to the more effective correlation between dipole orientation. As a consequence of this, we evaluated the interaction energy under the assumption of dipoles oscillating around only one of the two equivalent stable equilibrium positions, which are separated by a barrier [2–5], denoting by E that one of these regions which contains the origin of the expansion. Since the interaction potential energy between the dipoles was expanded around one of these equilibrium points, the configurational probability density cannot be represented adequately far from this point with a finite number of terms, for jaj ? 1, where a is the interaction parameter (see Eq. (20 )), because convergence is not uniform with respect to a in the semi-infinite interval 0 6 jaj < +1. However, asymptotic convergence of the series expansion of the integral was proved in [3], while in [4] it was proved that the configurational integral, extended to the region external to E, of a finite number of terms of the series, vanishes asymptotically in the limit a ? 1. Consequently, the integral extended to the whole space of the variable coordinates of the series expansion truncated after a finite number of terms converges to the true integral over E without altering its limiting value. This situation is substantially unchanged by the introduction of a uniform electric field, which orients the supermolecule preferably in the direction of the applied field, in order to minimize the electrostatic energy. Aim of this Let* Corresponding author. Fax: +39 010 3536199. E-mail address: [email protected] (V. Magnasco). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.07.064
ter is the evaluation of the average energy of the supermolecule in an electric field of strength e, so allowing for the computation of one-sided averages which are not attainable through symmetrical procedures. 2. Keesom integral in electrostatic field We proceed to the evaluation of Keesom integral, or configurational free energy, in a uniform electric field, assuming thermal equilibrium, for a supermolecule formed by two identical dipolar molecules A and B. It is introduced the potential energy functions V (X) and W (X), where X (#A,#B,/) is the set of three internal coordinates defining the mutual orientation of dipoles. It is assumed that the supermolecular axis R is parallel to the external field e, therefore
VðXÞ ¼ bkTðcos #A þ cos #B Þ b¼
le
ð10 Þ
kT
WðXÞ ¼ akTðsin #A sin #B cos / 2 cos #A cos #B Þ a¼
ð1Þ
l2
ð2Þ ð20 Þ
R3 kT
where / is the angle between the planes containing R and the dipoles lA, lB, respectively, and #A,#B are the polar angles of each dipole with respect to R. Therefore it is defined the integral over configuration space
Kða; bÞ ¼
Z
VðXÞ þ WðXÞ dX exp kT
dX ¼ sin #A sin #B d#A d#B d/
ð3Þ ð30 Þ
363
M. Battezzati, V. Magnasco / Chemical Physics Letters 496 (2010) 362–364
Lastly, we obtain the one-sided average in the metastable state as
3. Evaluation of K (a, b) The asymptotic evaluation of the Keesom integral (3) in the limit of large values of jaj was already accomplished in Refs. [2,4,5], a different approach being proposed in [6]. Our procedure consists in introducing vector planar coordinates #A, #B and in writing [4]:
d/A dX ¼ d#A d#B
sin #A sin #B #A #B
ð4Þ
where /A is a dummy variable denoting the rotational symmetry of the supermolecule around R as a whole, which contributes a factor 2p. Therefore we define the one-sided Keesom integral in the metastable state as
Kða; bÞ ¼
1 2p
ZZ
sin #A sin #B #A #B j#A jþj#B j6p VðXÞ þ WðXÞ exp kT d#A d#B
ð5Þ
We now expand the trigonometric functions in powers of #A ; #B , upon defining the normal coordinates a, b:
a ¼ #A þ #B
ð6Þ
b ¼ #A þ #B the Jacobian of the transformation being
@ða; bÞ ¼4 @ð#A ; #B Þ
ð60 Þ
ZZ
1 2 dadbqða; bÞ 1 ða þ b2 Þ þ h:o:t: 12 j#A jþj#B j6p ð7Þ
ð12Þ
which gives the energy correction at small values of the field strength. If the alignment is not complete it should be valuable to evaluate the average energy of the supermolecule in an electric field pointing in a direction oblique with respect to the axis R. If the deviation angle c is not too large, taking the plane containing R and the electric field e as reference plane containing the axis g, from which rotations /A and /B are measured, the Keesom integral is written
1 dadb exp 2ða þ bÞ þ ð3a þ bÞa2 4 all space 1 2p 2 þ ða þ bÞb þ cag 4 ð3a þ bÞða þ bÞ c2 exp 2ða þ bÞ þ h:o:t: 3a þ b
c¼
1 8p
ZZ
eg l
ð13Þ
ð130 Þ
kT
2
Kða; b; cÞ
ð8Þ
2pk T 2 R6
l2 ð3l þ eR3 Þðl þ eR3 Þ " 2 l2 el þ 3 exp kT
R
3l2 ec2
#
kTð3l þ eR3 Þ
ð14Þ
From Eq. (13) follows the average interaction energy with the oblique electric field:
4. Results Eq. (7) can be evaluated to the desired accuracy, yielding an expansion in inverse powers of 3a + b and a + b. It was proved in Ref. [4] that any sum of a finite number of terms of this expansion pffiffiffi converges exponentially to zero outside of a circle of radius p= 2. Furthermore, it was proved in [3] that the integral extended to the region satisfying j#Aj + j #Bj 6 p converges asymptotically to the Keesom integral inside of that region, which is therefore the limit of the integral extended to the whole space:
Kða; bÞ
2 hVðXÞi 2le 1 þ 3a
At small values of deviation angle c Eq. (13) yields, using Eqs. (10 ), (20 ), and (130 )
where q(a, b) is obtained by expansion of Eq. (3):
1 1 qða; bÞ ¼ exp 2ða þ bÞ þ ð3a þ bÞa2 þ ða þ bÞb2 4 4 12a þ b 4 4a þ b 4 b 2 2 b a ab 1 192 192 32 4a þ b j a bj2 þ h:o:t: 48
ð11Þ
Putting b = 0 into the bracket, we obtain:
Kða; b; cÞ
Then Eq. (5) is recast into the form
1 Kða; bÞ ¼ 8p
@ hVðXÞi ¼ bkT ln Kða; bÞ @b " 2 1 1 5a2 þ 4ab þ b þ þ le 2 þ 2 3a þ b a þ b 3ð3a þ bÞ ða þ bÞ2 # 2 2 63a2 þ 24ab þ b 7a2 þ 8ab þ b þ h:o:t: 6ð3a þ bÞ4 6ða þ bÞ4
2p exp½2ða þ bÞ ð3a þ bÞða þ bÞ " # 2a þ b 12a þ b 4a þ b 1þ þ h:o:t: 3ð3a þ bÞða þ bÞ 6ð3a þ bÞ2 6ða þ bÞ2
@ @ ln Kða; b; cÞ hVðX; /A Þi kT b þ c @b @c " # 2 2 c 1 1 2kTc kTb 2 þ þ þ þ h:o:t: 3a þ b 3a þ b a þ b 3a þ b
ð15Þ
In the limit of small correlation between dipole orientation (or a ? 0) Eq. (15) yields
" hVðX; /A Þi 2le1
1 1þ 2
2
2l j e j þ2kT
eg e1
kT
le1
# þ h:o:t: ð16Þ
Therefore, dipoles which are uncoupled orient preferably toward the field direction, being dis-aligned by thermal fluctuations.
ð9Þ As b ? 0, it may be ascertained that the rhs of Eq. (9) approaches the limiting value which we obtained in Refs. [2,4,5], while for a ? 0 it yields the correct asymptotic value
Kð0; bÞ
2p 2
b
expð2bÞ
ð10Þ
5. Conclusions Eq. (11) allows for the evaluation of the average energy of the supermolecule in the electric field of strength e, showing that the present calculation is suitable for computation of one-sided averages which are not attainable through a symmetrical procedure.
364
M. Battezzati, V. Magnasco / Chemical Physics Letters 496 (2010) 362–364
There has been done recently considerable effort in order to obtain adiabatically oriented molecules in thermal equilibrium [7] in a static electric field, either by ‘brute force’ orienting [8], or by femtosecond laser pulse and preparation of state-selected molecules using a hexapole [9]. Though the efficiency of the alignment is high, nonetheless it may be not complete, therefore calculations have been made including this case. It is pointed out, however, that our calculations refer to idealized experimental conditions, which may differ from the concrete situations resulting from laboratory practice. It is possible in this way to take account also of incomplete alignment and orientation, which appear to be produced in actual experiments, by evaluation of averages over the angular distribution of supermolecular axis around the field direction. Integration of the rhs of Eq. (14) over dc with the proper angular factor gives the configurational free energy taking into account all orientations of the supermolecular axis with respect to the field. Notice that the angular distribution (14) becomes increasingly flat with distance R,
so that dipoles which are far apart can occupy with equal probability all relative positions.
Acknowledgment Support by the University of Genoa is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
W.H. Keesom, Z. Phys. 22 (1921) 129. M. Battezzati, V. Magnasco, J. Phys. A: Math. Gen. 37 (2004) 9677. M. Battezzati, V. Magnasco, J. Phys. A: Math. Gen. 38 (2005) 6715. M. Battezzati, V. Magnasco, C R Chim. 12 (2009) 854. M. Battezzati, V. Magnasco, Chem. Phys. Lett. 492 (2010) 179. P.C. Abbott, J. Phys. A: Math. Theor. 40 (2007) 8599. H. Stapelfeldt, T. Seideman, Rev. Mod. Phys. 75 (2003) 543. H.J. Loesch, A. Remscheid, J. Chem. Phys. 93 (1990) 4779. O. Ghafur, A. Rouzée, A. Gijsbertsen, W. Kiu Siu, S. Stolte, M.J.J. Vrakking, Nat. Phys. 5 (2009) 289.