A new insight into the mechanism of intermolecular forces

Chemical Physics 171 ( 1993) l-7 North-Holland

A new insight into the mechanism of intermolecular forces E.A. Power

and T. Thinmamachandran

Departments of Mathematics and Chemistry, University College London. London WCIE 6BT, UK Received 15 October 1992

A generalized version of the response theory within molecular quantum electrodynamics is used to determine intermolecular interaction energies. The method brings into focus the important role of the energy density associated with the electric field. It is first applied to obtain the well-known London Re6 potential to illustrate the powerful nature of the method. For this case the appropriate energy density is entirely electrostatic. A straightforward generalization that includes contributions from dynamic fields leads to the fully retarded dispersion force. An extension of this approach to calculate intermolecular interactions involving molecules in excited states is given and a new power law with an R -* dependence is predicted.

1. Introduction In chemical physics the understanding of the nature of the forces between neutral molecules was one of the major successes of quantum mechanics [ l-3 1. London [ 1 ] showed that the intermolecular energy of a pair of molecules followed naturally from the application of quantum mechanics to the molecular pair as a composite system. He derived the famous R -6 law using second-order perturbation theory with the electrostatic coupling energy ( 1 ), in Gaussian units, V(R)=

~~,(A)“(B)(6,-3R,R,)

as the perturbation. --

l&

The energy shift was found to be

aP”(A)~~(A~~(B)P;O(B) mo no

which, for a randomly m=

(1)

2

c

3R6 m,n

oriented

(6,_3ff,JQ(~U_3R,Q) pair, becomes

lr”O(A)121r”o(B)12 E,,+E,

(2)

.

(3)

This result gives the London potential in terms of the transition dipole moments and not explicitly in terms of the ground state polarizabilities of the molecules. The fact that the energy denominator is the sum of transition energies of both molecules makes it impractical to express the molecular part of (3 ) as a product of static polarizabilities. However London showed, by defining an average energy E= IAZB/(IA + ZB) where IA and ZBare the ionization potentials, that AE( R ) can be approximated by

with CY*and CX~the static polarizabilities. Subsequent developments showed that the London result is valid for separations R smaller than the reduced wavelengths h/E,, associated with the relevant transitions. When the 0301-0104/93/$06.00

0 1993 Elsevier Science Publishers B.V. All rights reserved.

2

E.A. Power and T Thirunamachandran /Chemical Physics 171(1993)

I- 7

separations are comparable with, or larger than these wavelengths a quantum electrodynamical treatment is required. For two ground state molecules the complete dispersion potential was first found by Casimir and Polder [ 41. This potential has the required London expression for small R.On the other hand for large R the asymptotic form of the Casimir-Polder potential falls off as R -'and is [ 5 ] 23

aA@

hE(R)~-~fic~.

In this paper we develop a new approach to obtain intermolecular energy shifts. There are two steps in this method. First the field in the neighbourhood of a molecule is obtained. This field, for short intermolecular separations, is essentially electrostatic; however fully retarded fields are required for the general case. Secondly, the response of the second molecule to this field is given through the dependence on its dynamic polarizability at the transition frequencies of the source molecule. The novelty of the present approach is that the intermolecular interaction can be viewed as arising from the response of one molecule to the square of the field, and hence the electric energy density, of the other. In section 2 the new approach is applied to obtain the London potential with the aid of electrostatic energy densities. The full electrodynamical energy densities are used in section 3 to demonstrate the validity of the new method in the retarded regime. Finally we extend earlier work [ 6-8 ] by applying this response theory to the case of a pair of non-identical molecules where one is in an electronically excited state. In this situation the electric energy density has an additional contribution from the possibility of real photon emission. We show that a manifestation of this term is that the potential for large R is dominated by an R-‘-dependent part, and we give the general expression, a new result, for the intermolecular interaction energy.

2. London force deduced from electrostatic energy densities The electric energy density associated W(r)=

is

&E*(r),

which, for the electrostatic W(r)=

with an electric field E*(r)

field due to a dipole p at the origin, is

1

-81Cr6

h&(s,k

-

3F,Fk)
(7)

.

Throughout this paper we use Gaussian units. In quantum mechanics the analogue of (7) is the energy density associated with a molecular transition q+-p with moment p 4p. It is useful to define the field E r(qcpj(r)=

(8)

$#~(&-3~1~,),

and for a randomly

oriented

molecule the corresponding

energy density is (9)

For the calculation densities,

of the London

energy between

two molecules

in their ground

states we need the energy

E.A. Power and T. Thirunamachandran

/Chemical Physm 171 (I 993) I- 7

3

and

W’(B) I2

W,,(r) = 4xlr-RJj These densities from molecule A) through its n+O. Thus the

AE= - 4 1

(11)

16.

are then used in the determination of the mutual interaction energy. The energy density arising A influences molecule B. The latter responds to the energy density for the transition m+O (in dynamical polarizability at frequency o,,,~. Similarly molecule A responds to B for the transition total energy shift for the pair with isotropic polarizabilities cr*( o) and aB (0) is

aB(wmo)Efo (RB I- $1 a”(w,o)Eio (RA)

m

(12)

3

n

where cy( o) is the molecular

dynamic

polarizability

given by (13)

In eq. ( 12) E,,(r) is the electric field due to A associated with the me0 transition and E,,,(r) that from nt0 of B. The summation in ( 12) and ( 13) is to be interpreted to include integration over continuum levels. From (S), (lo), (11) and (13), with R=RB-RA, weobtain

Irno(B) I2 IP”(A)

Eno E2

M(R)=-;n;

=--

2x

IlO

_E2

I2

??I0

We(A) 121rno@) I2

3R 6 m,n

&,o +E,o

which is the London result. It has been found useful for computational using the result ab d” (a2+U2)(b2+U2)

(14)

’

(‘, “o)

purposes to express the energy shift in terms of polarizabilities.

’

By

(15)

0

the energy shift ( 3 ) can be converted into an integral over an imaginary frequency of dynamic polarizabilities ( 13) of the molecules [ 9, lo]. The resulting expression AR(R)=-~~a*(iu)c~~(iu)du.

(w = ck= icu) of the product for the energy shift is (16)

0

3. Retarded dispersion forces We have shown in section 2 that the formula ( 12)) which expresses the energy in terms of the dynamic polarizabilities and the electrostatic fields, leads to the R -’ London potential for a pair of ground state molecules. A remarkable feature of this formula is that it is applicable in more general circumstances. Firstly, the restriction to both molecules being in their ground states can be easily relaxed. This is discussed in section 4. Secondly, the physically transparent formula ( 12) holds even within the framework of quantum electrodynamics. A derivation of the energy shift for the molecule A in an electric field E in terms of the dynamic polarizability CX*is given in the Appendix. This together with its symmetrical counterpart gives ( 12 ) with E being made up of the vacuum and the source field as given below. In quantum electrodynamics the electromagnetic field is part of the total

4

E.A. Power and T Thirunamachandran

/ Chemrcal Physics I71 (I 993) I- 7

dynamical system [ 111 and has an energy density even in the absence of sources due to vacuum fluctuations [ 121. In the presence of a molecule the electric field operator in the Heisenberg picture can be written as the sum of two terms: E I (r >t)=E(O)(r I

2t)+Ep”=(r

3t) >

(17)

where E(O) is the vacuum field; in the electric dipole approximation qcpis [ 131

the source-dependent

term for the transition

(18) where the molecule is at the origin. It may be noted that in the near zone, with retardation completely ignored, the source-dependent field ( 18) reduces to the electrostatic expression (8). From ( 17) the electric energy density is W= ;

{ [E”‘(r,

t)12+ [E”‘(r,

t).E”““(r,

t) +E”““e(r,

t)-Eco)(r,

t)] + [E”““(r,(r, t)12} .

(19)

The first term in ( 19 ) corresponds to the vacuum fluctuation energy and energy shifts are calculated relative to this zero-point energy. However the vacuum field does indeed contribute to the energy shifts through the second term of ( 19). The third term of ( 19 ) is the quantum analogue of the classical expression (7 ). It is important to note that it does not give the total energy shift for all distances from the source. At very large distances, i.e. in the far zone, the energy density falls away as r- ‘. This asymptotic term comes entirely from the second term involving the vacuum field E (O). However in the near zone the r-’ contribution from the second term is zero so that the energy density in this regime is in fact the same as the electrostatic expression (7). To make a complete evaluation of the energy density ( 19 ) we use the Heisenberg equations for the molecular transition moments. These equations are used to calculate the value of the retarded moment in the expression for the electric field ( 18). The retarded electric field of the source and the vacuum electric field are both required to obtain the electric energy density. We have shown [ 141 that the expression for this energy density, correct to second order in the dipole moment, associated with the transition m +O may be written as an integral over imaginary frequency cu: W~(r)=~k-ICmo]2~duu6~~(~~2ur)(~+&+~+-$+&). 0

Substituting (20) for the energy density associated with molecule A and the corresponding of molecule B into the energy shift formula ( 12) we obtain

expression

for that

~,o~~oI~“O(B)121~mo(A)12 4 1 97czlcRl,n-

AE(R)=-

(Go-k2,o)

OD duu6exp(-2uR)

A---m0

0

duu6aA(iu)cuB(iu)

1 2 1 kio+u2 >( u2R2+i.%i3+ exp( -2uR)

- 5 + - ITi + - 3 U6R6 U4R4 u5R5

(21)

0

The energy shift is the well-known Casimir-Polder interaction energy [ 4,151. In the far zone (kR >> 1) the dynamic polarizabilities may be approximated by their static counterparts and (2 1) then reduces to the R -‘-

E.A. Power and T. Thirunamachandran

/Chemical Physu

dependent energy ( 5 ). In the near zone (kR K 1) the exponential leads to the integral expression (7) for the London potential.

171(1993)

I- 7

is set to unity and only the R -’ retained;

5

this

4. Interaction between two molecules with one in an excited state The approach through response theory to obtain energy shifts as used above is, with suitable modifications, applicable for systems involving excited molecules. For a pair of non-identical molecules A and B with A in its ground state and B in an excited state ]p) the energy shift is found from

(22) where (Y’ (0) is now the dynamic polarizability of B in the state 1p) and Em0( RB), Enp(RA) are the source fields from molecules A and B respectively. Clearly o,, in the second term, can take either sign depending on whether the transition ntp is upward or downward. There are physically significant contributions from downward transitions. These arise from the possibility of real photon emission and give an extra term in the electric field due to B in Enp(R). The additional term in the electric energy density is [ 141 3 I+&+k4 Pn Pn

r4

The corresponding

contribution

(23)

.

to the intermolecular

shift for an intermolecular

R is (24)

The remaining contributions and those arising from upward transitions 3. The total contributions from virtual transitions is

are the same as those found in section

00

- 5

I

duu6aA(iu)aB(iu;p)

exp( -2uR)

0

1 -+&T+&+&5+&i)~ u*R*

(25)

where aB(o; p) is the dynamic polarizability of molecule B in its excited state ]p). Thus the complete expression for the intermolecular interaction between molecule A in its ground state and B in excited state Ip) is the sumof (24) and (25). It is of interest to examine the asymptotic limits of this interaction and compare with those for a pair of molecules in their ground states. First we find the small R limit by retaining the R -’ terms in (24) and (25). The contribution from (24) associated with real photon emission is -4

I ”

a”(w,)

4lr;~~)l*

4 c ad~~~A)1*l~““(B)1* 3hcR6 ,,,,,, mO-WIlP EmcEp

EncEp

=-A

=

1

I~mo~~~121~np~~~12(o,,:,

Enrni;nE,

The virtual photon contribution

+ Pn

’ %lo -

(25 ) in this limit becomes

OPfl

).

(26)

6

E.A. Power and T Thirunamachandran

/Chemical

Physics 171(1993)

1-7

03

-5

I

ducrA(iU)aB(iU;p)=-&

1 w,l~“(A)l’o.,l~~~(B),2~ 0

=-~~~lPm"(~)121~~p(~)12(l,ol+l~~~

where we have used the integral this energy is

3frcR6

IN”%)

I’lr”(B) c%?lo+ &p

E.m;>p

(27)

ojq> nP

-Lx

(w~o+u2~l;wL+u2)

m,n

0

( 15 ). Partitioning

I2 +

2 3ficR6

nP

m

(27 ) with respect to upward and downward

c

IPoW)

Enm;nEp

Thus the total energy shift in the near zone obtained

121r”(B)

%?I0 +(%I

I2

transitions

of B

(28)

*

by adding (26) to (28) is (29)

We note that the double sum in (29) is unrestricted as is to be expected from elementary perturbation theory. In contrast to the result for a ground state pair, the denominator (E,o+Enp) in (29) can take either sign so that the London force between an excited molecule and a ground state molecule can be repulsive. Turning to the large R limit the asymptotic limit of (25 ) from virtual exchange is R -’ as in the dispersion case discussed in section 3. However the limit of (24) associated with real photon emission shows an R -2-dependence and is 2 u(R)-_-&

C

s:E,

aA(wp~)l~"p(~)12k~,=_

5 0

&

c

rn,R E”<&

wmOI/cmo(A)12W~~lpnp(B)12. 2

~nlo - 0,”

(30)

This novel energy shift can be important since it is much larger than the asymptotic Casimir R -’ potential when Ikl R x=-1. The molecular-dependent terms in (30) can be large when m+O and ntp transitions are nearly resonant; this is in contrast to (5 ) where there are no such resonances. We illustrate the relative importance of this shift by calculating the ratio of (30) to the Casimir-Polder potential ( 5 ). Within a two-level approximation this ratio is of the order of ( 1) (R/I) ‘7 where q is a measure of the near-resonance of the transitions in A and B, (31) Thus for Rx 1.51 and q z 2.5 the new predicted shift is an order of magnitude bigger than the Casimir-Polder potential. We remark that the response method used in this work is not applicable to the calculation of interactions between identical molecules involving exact resonance. In this case the absorption and emission can be real photon processes and the dominant contribution to the interaction energy is of lower order (linear in the dipole moment of each molecule). Finally we note that the concept of an intermolecular potential involving an excited molecule is a valid one provided the decay time is sufficiently long in comparison with the time taken for an electromagnetic signal to be transmitted between the molecules. For a lifetime of the order of lo-* s the notion of an intermolecular interaction energy is applicable up to all distances of physical interest.

Appendix In this appendix,

we derive the energy of a molecule in an electric field E in terms of its dynamic polarizability.

E.A. Power and T Thirunamachandran

The starting point for this derivation [ll]:

is the multipolar

/Chemical Physics 171(1993)

Hamiltonian

I- 7

(A. 1) in the electric dipole approximation

H=H,,,,,+H,,,-p-E(R). This Hamiltonian H

=ei’;He-‘s new

I

(A.1) is transformed

by the unitary

transformation

3

(A-2)

with Schosen so that the -p-E term is eliminated moments, the transformed Hamiltonian is

H,,=H,,+H,d+i[S,H,,l+H,dl-~.E(R)+~ = H,,,ol + Hrad - L; IX

(except on the energy shell). To second order in the transition

[S, [SfLol+Hmdll-i[S,

--P-E(R)I

-P.E(R)I

(A.3)

(A-4)

= Hmol+ Hrad + Hint 7

(A.5)

where S is found from i[S,H,Or+&d]=P’E(R).

(‘4.6)

In terms of the base states 1E,)

for the molecule and I N(k, A) ) for the photons, (A-7)

With this choice, the diagonal element of Hi,, in (A.5 ) is --

1 p”“~E(R),u”‘“~E(R) 2 E:, - (fio_~)~

(‘4.8)

’

which, for a molecule with isotropic polarizability

(Y(0) is

-&x(o)E2(R).

References [ 1 ] F. London, Z. Physik 63 ( 1930) 245. [2] H. Margenau, Rev. Mod. Phys. 11 (1939) 1. [ 31 J.O. Hirschfelder, Intermolecular forces (Wiley, New York, 1967). [4] H.B.G. Casimir and D. Polder, Phys. Rev. 73 ( 1948) 360. [5] H.B.G. Casimir, Proc. KoninkJ. Ned. Akad. Wetenschap. 60 (1948) 793. [6] R.R. McLone and E.A. Power, Proc. Roy. Sot. A 286 ( 1965) 573. [7] L. Gomberoff, R.R. McLone and E.A. Power, J. Chem. Phys. (1966) 4148. [8] M.R. Philpott, Proc. Phys. Sot. 87 (1966) 619. [ 91 A. Dalgamo, in: Quantum theory, Vol. 1, ed. D.R. Bates (Academic Press, New York, 196 1). [lo] H. Margenau and N.R. Kestner, Theory of intermolecular forces (Pergamon Press, Oxford, 1969) p. 5 1. [ 111 D.P. Craig and T. Thirunamachandran, Molecular quantum electrodynamics (Academic Press, New York, 1984). [ 121 G. Compagno, R. Passante and F. Persico, Phys. Rev. A 31 ( 1985) 2827. [ 13 ] D.P. Craig and T. Thirunamachandran, Chem. Phys. I35 ( 1989) 37. [ 141 E.A. Power and T. Thirunamachandran, Phys. Rev. A 45 ( 1992) 54. [ 151 C. Mavroyannis and M.J. Stephen, Mol. Phys. 5 ( 1962) 629.

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