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Very-long-range static dipole moment of two coupled systems
CHEMICAL
Volume 98, number 4
VERY-LONG-RANGE
J.M. VIGOUREUX
1 July 1983
PHYSICS LETTERS
STATIC DIPOLE MOMENT OF TWO COUPLED
SYSTEMS
and L. GALATRY
Laboratoire de Physique hfoIhlaire*. FacuIt.2 des Sciences et des Techniques. Uuiversite’ de Besungon. La BouIoie, 25030 Besancon Cedex. France Rcccivcd
14 March 1983; in final form 28 April 1983
The expression for the very-lon_g-range dipofe moment of a pair of non-identicai centrosymmetric systems (varying as Re8) is obtained. Our expression isgiven in terms of poIarisabiitie.s and hyperpoiarisabilities of the two systems and is compared !\ith t\\o previous results.
I, Introduction The existence of the long-range static dipole I~lo~~lent of two spheric&y symmetric systems was pointed out by Buckingham [ 1J _The theory was further developed by Brown and Whisnant [2] and Hunt [3] _Recently, Galatry and Gharbi [4] gave a more physical form to these earlier results by expressing them in terms of dynamical polarisabilities and hyperpolarisabihties of the two systems. In a more recent paper Craig and Thirunamachandran [S] obtained these same results by using quantum electrodynamics. However ail these calculations are performed in the London limit. We recently presented a general study giving the dipole moment of a pair of non-identical systems at any distance [6f _The limit form of our generaI expression Ieads in the near zone (London) to the same result 3s the one of Galat~ and Gharbi [4] and Craig and T~run~mach~dran 151. In the far zone (Casimir) we find an R -8 dependence of the dipole moment, in harmony with expectations of Craig [7] but at odds with a previous proper of one of us [S] _Our wish in this letter is therefore to discuss these two conflicting results. AIthough the cslculation of ref. [6] applies to ail distances, we consequently onIy present here the limiting case of \ery long distances.
2. Theory
The dipole moment id of a p&r of mole&es can be obtained pled system wbcn an external static electric field Es is applied P&- = --p-E,
+ _.. or
by derivation
of the energy shift AE of the cou-
g’i = -i3(AE)/aEsi _
(1)
To find p, our first step is to calculate the R-dependent part of Af? (R is the distance between the two systems). To do this we use quantum electrodynamics in dipole and quadrupole approximation. In this formalism, interactions are mediated by virtual photons. In the case when the two coupled systems are atoms or centrosymmetric molecules, the parity selection rules show that AE can be obtained in the lowest order from graphs of fig. 1 and their reflections (we do not represent there the additional graphs which onIy differ by the possible different orders of quadrupole 0 and dipole 0 vertices). In these diagrams, the horizontal line represents the external static fieId * Associated with the Centre National de Ia Recherche Scientifique. 324
0 009-2614/83/0000-0000/S
03.00 0 I983 North-Holland
CHEMICAL
Volume 98, number 4
1 July 1983
PHYSICS LE’ITERS
r
cj A
B
A
B
la
A
B
r I
le
IC
w+dw I
A
Ii
B
A
B
A
If
Id
lb
A
+
Ef Fig. 2.
:
EF
Illustration of notation usedin the calculation_
Fig. 1. Graphs corresponding to two virtual photon eschanges in the presence of a static external field. Symbols o and o in-
dicate a dipoiar and a quadrupolarinteraction respectively_
and each wave line represents a virtual photon. The Feynman propagator of such a photon can be written (o = clkj): DrF(~Afl, r&)
= - 5 [inc/.~o(2n)3 I s
dk exp [-ick(~~ - .rl)]g~(lkl,R)
-_
,
(2)
with @lkl,R)
= exp(ilklR)
W3 ~v(l~l,R)
(3)
and (4) We do not delve into justifications of these expressions since it is lucidly set forth in electrodynamic books and papers. Using notation presented in fig. 2 (states I-, s and p are the virtual states of A and B respectively; letters i,j, k, I, m, n stands for Cartesian components of quadrupole and dipole moments; the energy of the initial and of the final states of A and B is Ef), 6” = e(pmpIrklqmq) is the dipole transition moment, and Oft?= e
G l
PG, ,
(/C;:
+E;Yh;J
fP
(5)
isthe dynamical polarisability of B, and
l +lick
+Efr - Ack
(6) 325
Volume 98, number 4
1 July 1983
CHEMICAL PHYSICS LETTERS
is the dynamical hyperpolarisability
of A, the static dipole moment
of the two systems is then (k = in):
3. Far-zone limit The definition (4) of Ir,,, and irii being quite general, our result (6) is valid at any distance. However, as explained above. we restrict ourselves in this paper to the only case of very long range. Noting Er =fick, the charActeristrc energy of transitions in the coupled system, the ‘&far-zone limit” is usually defined by the condition X-,R $ 1. Because of the quickly decreasing factor exp(--2uR) the integrand of (7) quickly decreases outside the interval 0 < u < some l/R < k,. Besides by looking at (5) and (6) we can easily see that inside this same interval afizr) = a(O) and x(iu. -iu) z x(0,0) are practically constant_ In fact, the condition 0
&I s
0 Takings 1 0 we obtain
esp(--?uR)
(a + pU + -&
+ 822 -f a4 + ,$‘d) .
(8)
= uR and using do _vylJehp(--2v)
mn6ij
= ?2!/3’r+ l
-
1896ii-
(9)
R&,1 R2
Ri~j~ + Rj~ii R
RiRj RiRjRm R, + 693 R’ R4
- 3156,,
11 .
(10)
4. Particular case of two spherical systems In this case, it is possible to perform summations and (6) we have aJJI
i = %2X6 mI-
and, in a quite similar way [9]
326
on the magnetic quantum numbers ma, ml
and nzn;j_In (5)
(11)
CHEMICAL PHYSICS LETTERS
Volume 98, number 4
xn Zkj =
xxxxx-
fs,,skj
+ ~snks~ +
1 July 1983
(12)
~6nj6,
be underlined that, according to ref. [9] a,, = ;app and xXxXx = &&Crscra, where the convention in which a repeated Greek subscript denotes a summation over all three cartesian components has been used)_ Using (11) and (12), the exact expression for the static dipole moment is (it must
a+&
R s
- h ew--2uR)
z2 C&$Xxzxxx -~x~xxx)(l+&+-&+-&+~4+-&)
0 and in the far-zone limit:
.i,e~.-,R,(~+~~~~3~+~+~)
(13)
(14)
0 After integrating over u, we find
(IS)
5. Discussion Eqs. (10) and (15) clearly show that in the far-zone limit, the static dipole moment of the two systems varies as Re8. This result agrees with the expectation of Craig [7] but not with the paper of Galatry [S] _In fact, in this latter work, p is seen to vary as Rd2 (although typewriting errors indicate RA3)_ The reason for this important difference can be found in the fact that in the dassicaI field of the linear electric dipole (eq. (1) of ref. [8]) EBp(w) = m*,(w)
/Q(o,
R) exp(--iwR/c)
(16)
there appears a factor exp(-ioR/c), instead of the exp(-ilolR/c) of the Feynman propagator used here [eq. (3)] _ This difference does explain our two opposing results when (as in ref. [S]) integration is performed in the interval -m to m. The use of (3) leads to (14) where the integrand contains the factor exp(--2uR), which obviously leads to the R-* variation ofu. However introduction of (16) in the field term E(w) ~,!?(-a) (eq_ (2) of ref. [S]) leads to an expression similar to (14) but where the exponential factor is absent: +4u4 R3
+g
+ 14u3+31uZ+36u R4
R5
R=
R7 > -
(17)
At very long range this expression reduces to the only term u5R- 2 _Unhappily, the disappearance of the exponential factor makes the integral (17) diverge and this result has no sense. This comparison of these two methods of calculating shows that the use of the classical field of the linear electric dipole in molecular interactions may sometimes lead to ticklish problems_ A good way to conduct the semi-classical calculation is that of Casimir [lo] who takes the average of the advanced and the retarded dipole field. Underlining that a hertzian dipole is generally damped out by outgoing radiation, Langbein [l l] used the same method when he considered that the dipole of interest is surrounded by a large perfectly reflecting cavity to balance the outgoing waves and to obtain the energy back from infiiity. Let us note that this method of Casimir [lo] or Langbein [ll] was also used by Wheeler and Feynman [12] to subtract our divergent mass integrals in the theory of radiation.
327
Volume
98, number 4
CHEMICAL
PHYSICS
LElTERS
1 July 1983
6. Conclusion We show that at very long range, the static dipole moment of two coupled centrosymmetric systems varies as R-8. It only depends on the static polarisability and hyperpolarisability of the two systems. This can be clearly understood when the field emitted by A arrives at B, the dipole A has significantly fluctuated and “forgotten” its initial state. Motions of A and B are consequently uncorrelated. This can explain the factorisation of the polarisabihty and of the hyperpolarisabihty of A and B in eqs. (10) and (15). Moreover, only frequencies below c/R being important in interactions, we can easily understand that in the very long range our results only depend on very low frequencies and consequently on the static polarisability and hyperpolarisabihty u(O) and x(0,0). The comparison of quantum electrodynamics and semi-classical calculations shows that cautions needs to be taken in the latter to avoid various traps which are present in the use of electric dipole (or multipole) radiation.
References [ 1 ] AD. Buckingham, Propnet& optiques et acoustiques des fluides cornprimes et actions intermoliculaires [2j 1k.B. Broirn and D.M. Whisnant, Mol. Phys. 25 (1973) 1385. 13) K.L.C. Hunt, Chem. Phys. Letters 70 (1980) 336_ [4 ] L. tialatry and T. Gharbi. Chem. Phys. Letters 75 (1980) 427. [S] D P. Craig .md T. Thirunamachandran. Chem. Phys. Letters 80 (1981) 14. [6] JAI. Vigoureux, J. Chem. Phys., to be published. [7] D.P. Cr&g, private communication. [S] L. Galatry, Comet. Rend. Acad. Sci. (Paris) 291 (1980) 113. 191 4.D. Buckingham. Advan. Chem. Phys 12 (1967) 107. lo] II B.C. Casimir, J. Chim Phys. 46 (1949) 407. ,I1 ] D. Lanpbrin, Theory of van der Waak attraction (Springer, Berlin, 1974) p_ 72121 J.A. Wheeler and R-P. I~eynman, Rev. Mod. Phys. 17 (1945) 157.
328
(CNRS, Paris. 1959).
Volume 98, number 4
VERY-LONG-RANGE
J.M. VIGOUREUX
1 July 1983
PHYSICS LETTERS
STATIC DIPOLE MOMENT OF TWO COUPLED
SYSTEMS
and L. GALATRY
Laboratoire de Physique hfoIhlaire*. FacuIt.2 des Sciences et des Techniques. Uuiversite’ de Besungon. La BouIoie, 25030 Besancon Cedex. France Rcccivcd
14 March 1983; in final form 28 April 1983
The expression for the very-lon_g-range dipofe moment of a pair of non-identicai centrosymmetric systems (varying as Re8) is obtained. Our expression isgiven in terms of poIarisabiitie.s and hyperpoiarisabilities of the two systems and is compared !\ith t\\o previous results.
I, Introduction The existence of the long-range static dipole I~lo~~lent of two spheric&y symmetric systems was pointed out by Buckingham [ 1J _The theory was further developed by Brown and Whisnant [2] and Hunt [3] _Recently, Galatry and Gharbi [4] gave a more physical form to these earlier results by expressing them in terms of dynamical polarisabilities and hyperpolarisabihties of the two systems. In a more recent paper Craig and Thirunamachandran [S] obtained these same results by using quantum electrodynamics. However ail these calculations are performed in the London limit. We recently presented a general study giving the dipole moment of a pair of non-identical systems at any distance [6f _The limit form of our generaI expression Ieads in the near zone (London) to the same result 3s the one of Galat~ and Gharbi [4] and Craig and T~run~mach~dran 151. In the far zone (Casimir) we find an R -8 dependence of the dipole moment, in harmony with expectations of Craig [7] but at odds with a previous proper of one of us [S] _Our wish in this letter is therefore to discuss these two conflicting results. AIthough the cslculation of ref. [6] applies to ail distances, we consequently onIy present here the limiting case of \ery long distances.
2. Theory
The dipole moment id of a p&r of mole&es can be obtained pled system wbcn an external static electric field Es is applied P&- = --p-E,
+ _.. or
by derivation
of the energy shift AE of the cou-
g’i = -i3(AE)/aEsi _
(1)
To find p, our first step is to calculate the R-dependent part of Af? (R is the distance between the two systems). To do this we use quantum electrodynamics in dipole and quadrupole approximation. In this formalism, interactions are mediated by virtual photons. In the case when the two coupled systems are atoms or centrosymmetric molecules, the parity selection rules show that AE can be obtained in the lowest order from graphs of fig. 1 and their reflections (we do not represent there the additional graphs which onIy differ by the possible different orders of quadrupole 0 and dipole 0 vertices). In these diagrams, the horizontal line represents the external static fieId * Associated with the Centre National de Ia Recherche Scientifique. 324
0 009-2614/83/0000-0000/S
03.00 0 I983 North-Holland
CHEMICAL
Volume 98, number 4
1 July 1983
PHYSICS LE’ITERS
r
cj A
B
A
B
la
A
B
r I
le
IC
w+dw I
A
Ii
B
A
B
A
If
Id
lb
A
+
Ef Fig. 2.
:
EF
Illustration of notation usedin the calculation_
Fig. 1. Graphs corresponding to two virtual photon eschanges in the presence of a static external field. Symbols o and o in-
dicate a dipoiar and a quadrupolarinteraction respectively_
and each wave line represents a virtual photon. The Feynman propagator of such a photon can be written (o = clkj): DrF(~Afl, r&)
= - 5 [inc/.~o(2n)3 I s
dk exp [-ick(~~ - .rl)]g~(lkl,R)
-_
,
(2)
with @lkl,R)
= exp(ilklR)
W3 ~v(l~l,R)
(3)
and (4) We do not delve into justifications of these expressions since it is lucidly set forth in electrodynamic books and papers. Using notation presented in fig. 2 (states I-, s and p are the virtual states of A and B respectively; letters i,j, k, I, m, n stands for Cartesian components of quadrupole and dipole moments; the energy of the initial and of the final states of A and B is Ef), 6” = e(pmpIrklqmq) is the dipole transition moment, and Oft?= e
G l
PG, ,
(/C;:
+E;Yh;J
fP
(5)
isthe dynamical polarisability of B, and
l +lick
+Efr - Ack
(6) 325
Volume 98, number 4
1 July 1983
CHEMICAL PHYSICS LETTERS
is the dynamical hyperpolarisability
of A, the static dipole moment
of the two systems is then (k = in):
3. Far-zone limit The definition (4) of Ir,,, and irii being quite general, our result (6) is valid at any distance. However, as explained above. we restrict ourselves in this paper to the only case of very long range. Noting Er =fick, the charActeristrc energy of transitions in the coupled system, the ‘&far-zone limit” is usually defined by the condition X-,R $ 1. Because of the quickly decreasing factor exp(--2uR) the integrand of (7) quickly decreases outside the interval 0 < u < some l/R < k,. Besides by looking at (5) and (6) we can easily see that inside this same interval afizr) = a(O) and x(iu. -iu) z x(0,0) are practically constant_ In fact, the condition 0
&I s
0 Takings 1 0 we obtain
esp(--?uR)
(a + pU + -&
+ 822 -f a4 + ,$‘d) .
(8)
= uR and using do _vylJehp(--2v)
mn6ij
= ?2!/3’r+ l
-
1896ii-
(9)
R&,1 R2
Ri~j~ + Rj~ii R
RiRj RiRjRm R, + 693 R’ R4
- 3156,,
11 .
(10)
4. Particular case of two spherical systems In this case, it is possible to perform summations and (6) we have aJJI
i = %2X6 mI-
and, in a quite similar way [9]
326
on the magnetic quantum numbers ma, ml
and nzn;j_In (5)
(11)
CHEMICAL PHYSICS LETTERS
Volume 98, number 4
xn Zkj =
xxxxx-
fs,,skj
+ ~snks~ +
1 July 1983
(12)
~6nj6,
be underlined that, according to ref. [9] a,, = ;app and xXxXx = &&Crscra, where the convention in which a repeated Greek subscript denotes a summation over all three cartesian components has been used)_ Using (11) and (12), the exact expression for the static dipole moment is (it must
a+&
R s
- h ew--2uR)
z2 C&$Xxzxxx -~x~xxx)(l+&+-&+-&+~4+-&)
0 and in the far-zone limit:
.i,e~.-,R,(~+~~~~3~+~+~)
(13)
(14)
0 After integrating over u, we find
(IS)
5. Discussion Eqs. (10) and (15) clearly show that in the far-zone limit, the static dipole moment of the two systems varies as Re8. This result agrees with the expectation of Craig [7] but not with the paper of Galatry [S] _In fact, in this latter work, p is seen to vary as Rd2 (although typewriting errors indicate RA3)_ The reason for this important difference can be found in the fact that in the dassicaI field of the linear electric dipole (eq. (1) of ref. [8]) EBp(w) = m*,(w)
/Q(o,
R) exp(--iwR/c)
(16)
there appears a factor exp(-ioR/c), instead of the exp(-ilolR/c) of the Feynman propagator used here [eq. (3)] _ This difference does explain our two opposing results when (as in ref. [S]) integration is performed in the interval -m to m. The use of (3) leads to (14) where the integrand contains the factor exp(--2uR), which obviously leads to the R-* variation ofu. However introduction of (16) in the field term E(w) ~,!?(-a) (eq_ (2) of ref. [S]) leads to an expression similar to (14) but where the exponential factor is absent: +4u4 R3
+g
+ 14u3+31uZ+36u R4
R5
R=
R7 > -
(17)
At very long range this expression reduces to the only term u5R- 2 _Unhappily, the disappearance of the exponential factor makes the integral (17) diverge and this result has no sense. This comparison of these two methods of calculating shows that the use of the classical field of the linear electric dipole in molecular interactions may sometimes lead to ticklish problems_ A good way to conduct the semi-classical calculation is that of Casimir [lo] who takes the average of the advanced and the retarded dipole field. Underlining that a hertzian dipole is generally damped out by outgoing radiation, Langbein [l l] used the same method when he considered that the dipole of interest is surrounded by a large perfectly reflecting cavity to balance the outgoing waves and to obtain the energy back from infiiity. Let us note that this method of Casimir [lo] or Langbein [ll] was also used by Wheeler and Feynman [12] to subtract our divergent mass integrals in the theory of radiation.
327
Volume
98, number 4
CHEMICAL
PHYSICS
LElTERS
1 July 1983
6. Conclusion We show that at very long range, the static dipole moment of two coupled centrosymmetric systems varies as R-8. It only depends on the static polarisability and hyperpolarisability of the two systems. This can be clearly understood when the field emitted by A arrives at B, the dipole A has significantly fluctuated and “forgotten” its initial state. Motions of A and B are consequently uncorrelated. This can explain the factorisation of the polarisabihty and of the hyperpolarisabihty of A and B in eqs. (10) and (15). Moreover, only frequencies below c/R being important in interactions, we can easily understand that in the very long range our results only depend on very low frequencies and consequently on the static polarisability and hyperpolarisabihty u(O) and x(0,0). The comparison of quantum electrodynamics and semi-classical calculations shows that cautions needs to be taken in the latter to avoid various traps which are present in the use of electric dipole (or multipole) radiation.
References [ 1 ] AD. Buckingham, Propnet& optiques et acoustiques des fluides cornprimes et actions intermoliculaires [2j 1k.B. Broirn and D.M. Whisnant, Mol. Phys. 25 (1973) 1385. 13) K.L.C. Hunt, Chem. Phys. Letters 70 (1980) 336_ [4 ] L. tialatry and T. Gharbi. Chem. Phys. Letters 75 (1980) 427. [S] D P. Craig .md T. Thirunamachandran. Chem. Phys. Letters 80 (1981) 14. [6] JAI. Vigoureux, J. Chem. Phys., to be published. [7] D.P. Cr&g, private communication. [S] L. Galatry, Comet. Rend. Acad. Sci. (Paris) 291 (1980) 113. 191 4.D. Buckingham. Advan. Chem. Phys 12 (1967) 107. lo] II B.C. Casimir, J. Chim Phys. 46 (1949) 407. ,I1 ] D. Lanpbrin, Theory of van der Waak attraction (Springer, Berlin, 1974) p_ 72121 J.A. Wheeler and R-P. I~eynman, Rev. Mod. Phys. 17 (1945) 157.
328
(CNRS, Paris. 1959).