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On separation theorems for van der Waals interactions
CHEMICAL
Volume 97, number 1
ON SEPARATION A.C. TANNER,
THEOREMS
6 hiay 1983
PHYSICS LETTERS
FOR VAN DER WAALS INTERACTIONS
Ajit J. THAKKAR
Guelph-li’oterloo Centre for Graduate Work in Chemistry, Universiw of Waterloo. Waterloo. Ontario. Gmnda N2L 3Gl
and B. LINDER Department of Qzemistry. Florida State University. Takhassee. Rorida 32306. US.4 Received 24 January
1983
Two integral representations for energy denominators, the Casimir-Polder form and a convolution form, are used to obtain separation theorems for van der \k’aalsinteractions_ The separation theorems are shown to be different and complcmentary; the convolution form yields the total second-order energy while the Casimir-Polder form yields the dispersion ener_ey_
1. Introduction When the interaction of two systems a and b in nondegenerate ground states is treated by perturbation theory, the following expression arises for the secondorder interaction energy [ I] :
energies of system a are denoted by In) and 0,. while those of b are denoted by IN) and QN_ The quantities won = wo - w,, and R, = !$ - SL, are the negatives of excitation energies_ Af?) contains the induction energy U&i given by
&2)
=
xnd
(1)
g
Ir#O
IcoolFYlnO>12 $ IcooIrulo~)12 $
WOn
as well as the dispersion
;V#O
n0N
energy A!?$:? given by
where
0)
simply represents the Coulomb interaction between the particles of system a (with charges Zj and coordinates ri relative to an origin in a) and the particles of system b (with charges ZJ and coordinates R J relative to an origin in b); D is the vector between the two origins. The prime on the generalized summations signifies that the term with N = tz = 0 is not included_ The states and 0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
in eqs. (I) and (2), Troperties of both systems occur in every term; thus when the identities of a and b change, a great deal of labor is required to compute the new aE. A separation theorem expresses U as an integral which involves a product of quantities depending only on the separate systems a and b and thus rcduces the labor required to calculate Ak’ for all pairs of systems from a given set. 37
CHEMICAL PHYSICS LETl-ERS
97. number 1
Volume
Many years ago, Casimir and Polder [Z] introduced a now widely-used identity for the ener,T denominators of eq. (Zb):
so that the matrix as
:
n,,, = -
k
i
‘z)
-
theorems
%”
=_-
+ %,
1 2ni I
c+i-
c-i-
dz (OOn - =)(.Q20,v + =) ’
qs2
~,(--p)IN)(JtI
exp(-iq
-D)
~,(~)lo,
w q.j(-q)IOA
(6)
(3
=
F
=
TZJ exp(iq
ii
exp(iq -$
(2=)3’4 VB(q) Substitution
of eqs. (5) and (6) in (1) yields c+i-
tic21
-RJ)_
= ire2 J
dz
jep-z exp(-ip-D)
c-ix J-d4
z
exp(-iq
f2
-D)
(01 ~,(P)h)(J4
n
WOIl
y&)10) -z
(7) Note that in eq. (7) the term with N = JZ = 0 is included whereas it was explicitly excluded in eq. (1). This is possible because the term gives rise to a double pole with zero residue and hence its contribution to the contour integraI is zero. Eq. (7) can be written more compactly with the help of the response functions
potential and
-rj+Di
=$Jdq
- D>sdq
and
X
with c < - QoN_ Neht , following Bethe [4] _ the Coulomb is represented by the Fourier integral IiiR,
x
~A(p)~JZ)(oI
C4) (27r)3/4 VA(q)
The convolution form of the separation theorem can be derived as follows. First the contour in eq. (4) is deformed so that it begins at c - i- where c < --R,. The contour goes to c + im and is then complcicd b) a large semi-cicle in the left half of the complex plane. The contribution from the semi-circle goes to zero as its radius goes to infinity_ Thus 1
in eq. (1) can be expressed
where dz (Won - Z)(q)N
The symbol above the integral sign in eq. (4) indicates that the contour encloses whl in the positive sense but does not enclose the pole at -ROX_ The separation rheorem based on eq. (4) will be referred to as the consolution form. It does not appear to be widely recognized that the two integral representations lead to two different separation theorems_ The purpose of this note is to show that the two are complementary in the sense that the Casimi-Polder form yields the dispersion energy whereas the convolution foml yields the total second-order interaction energy.
2. Separation
$Jc.lp pm2exp(-ip
x (01
The separation theorem based on eq. (3) will be referred IO as the Casimir-Polder form. Another integral representation for the energy denominators [3 ] is
Won
elements
Itool Wl?LWl’ =
(“&I)
6 May 1983
exp[-iq*(RJ
- ri +D)]qm2,
B,@,w)=~
iv
(01 q&J)INxNI ~,(q)lO) R ON- +z
WI
CHEMICAL
Volume 97, number 1
6 h&y 1983
PHYSICS LETTERS
Thus the convolution form of the separation theorem for the total second-order interaction energy can be written as &?)
= ins2 c-G=
xs
c-i-
FLJ
dp dq exp I-*-
@ + 411~-~4-~
(9)
x LQUz,~)~:GLW).
Additional equivalent expressions can be obtained by replacingA __@,q?)B,*(q,p,z) by B_@,q,zW&,p,z) or the symmetrized form
in which z&j and @ are the nuclear and electronic parts of VA and the states In) are now pure electronic states of a; the response functions of b are treated analogously_ Eq. (9) can now be written as MC21 = inm2
c+im X J c-i-
A derivation similar in spirit to the above but starting with eq_ (2b) and using the identity (3) yields the Casirnir-Polder separation theorem for the dispersion energy:
&sJQ
a4 exp[--iD*@+q)]p-2q-2
+A’d’~,q,z)B,@‘*(q,P,z) +A(d’(p,q.s)~~)*(q,~,z)l -
&$-$ = (2n2i)-' X J+
&JJdp
where the
dq exp[--iD-
>
(13
fast two terms are the induction terms and
the last term is the dispersion term. Similarly the response functions A and B* In the Casimir-Polder formula (10) can be replaced by Acd) and Bed)* respectively
(P + 4)1F24-2
where A (a) = A$i) + A 5) and B(d) = B,‘d’ + BF)_
x awzrZ)~*(qrPJ),
(10)
where A(p,q,z) = A_@,q,z) + A+(p,q,z) and similarly for B. Eq. (9) can be put in a more familiar form by observing that the states In> and IN) are combined nuclear -electronic states. When the nuclear coordinates are frozen during the determination of the intermolecular potentials, only the ground nuclear states can produce non-vanishing matrix elements of the nuclear coordinates and these will be ordinary (c-) numbers. Thus the response functions of eq. (8) can be decomposed into two terms: A,&,q,s)=A,@)@,q,z)
+--$d)@,q>z),
where Asp’(p,q,z) x [v?)(q)
= [v:‘(p)
+ (0 l~~%p)lO>]
i- (0 1l&q)1 O)] I(2 z),
(114
3. Concluding remarks Clearly the convolution form or the separation theorem is not equivalent to the Casimir-Polder form. On the contrary the two are complementary since the convolution form can be used to obtain the total secondorder interaction energy which consists of the induction and dispersion energies, and the Casimir-Polder form can be used to obtain the dispersion energy separately_ It is well known [S] that if the spectral forms (8) are not convenient, then the response functions can be obtained by the variational solution of first-order perturbation equations which describe the interactions of the separate systems with an electromagnetic plane wave. If desired the perturbation IVcan be expanded in multipoles to obtain separation theorems for the van der Waals coefficients in terms of dynamic multipole polarizabilities When the multipole expansion is employed for interactions between pairs of neutral closed-shell 39
Volume 97. number 1
CHEMICAL PHYSICS LETTERS
atoms the two separation theorems yield identical results because in such cases the induction energy is solely
and in part by the United States National Health Grant No. GM23223.
made up of exponentially decaying terms whose asymptotic expansion is identically zero. Finally we note that the present work can be generalized to exchange pertur-
References
bation theories and perturbative calculations of correlation energies in the manner demonstrated by Langhoff 151 for the Casimir-Polder separation theorem.
Acknowledgement This work was supported in part by the Natural and Engineering Research Council of Canada
Scicnccs
40
6 May 1983
Institutes
of
[ l] J-0. Hirschfelder and W.J. Meath. Advan. Chem. Phys. 12 (1967) 36. [2] H.B.G. Casimir and D. Polder. Phys. Rev. 73 (1948) 360. [ 31 R. Fulton, unpublished; B.J. Lauren& J. Chem. Phys. 50 (1969) 3576: Chem. Phys. Letters 5 (1970) 3: J. Chem. Phys. 56 (1972) 619: R. Boehm and R. Yaris. J. Chem. Phys. 55 (1971) 2620. [4] H.A. Bethe, Ann. Physik 5 (1930) 325. [ 5 ] P-W. Langhoff, Chem. Phys. Letters 20 (1973) 33.
Volume 97, number 1
ON SEPARATION A.C. TANNER,
THEOREMS
6 hiay 1983
PHYSICS LETTERS
FOR VAN DER WAALS INTERACTIONS
Ajit J. THAKKAR
Guelph-li’oterloo Centre for Graduate Work in Chemistry, Universiw of Waterloo. Waterloo. Ontario. Gmnda N2L 3Gl
and B. LINDER Department of Qzemistry. Florida State University. Takhassee. Rorida 32306. US.4 Received 24 January
1983
Two integral representations for energy denominators, the Casimir-Polder form and a convolution form, are used to obtain separation theorems for van der \k’aalsinteractions_ The separation theorems are shown to be different and complcmentary; the convolution form yields the total second-order energy while the Casimir-Polder form yields the dispersion ener_ey_
1. Introduction When the interaction of two systems a and b in nondegenerate ground states is treated by perturbation theory, the following expression arises for the secondorder interaction energy [ I] :
energies of system a are denoted by In) and 0,. while those of b are denoted by IN) and QN_ The quantities won = wo - w,, and R, = !$ - SL, are the negatives of excitation energies_ Af?) contains the induction energy U&i given by
&2)
=
xnd
(1)
g
Ir#O
IcoolFYlnO>12 $ IcooIrulo~)12 $
WOn
as well as the dispersion
;V#O
n0N
energy A!?$:? given by
where
0)
simply represents the Coulomb interaction between the particles of system a (with charges Zj and coordinates ri relative to an origin in a) and the particles of system b (with charges ZJ and coordinates R J relative to an origin in b); D is the vector between the two origins. The prime on the generalized summations signifies that the term with N = tz = 0 is not included_ The states and 0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
in eqs. (I) and (2), Troperties of both systems occur in every term; thus when the identities of a and b change, a great deal of labor is required to compute the new aE. A separation theorem expresses U as an integral which involves a product of quantities depending only on the separate systems a and b and thus rcduces the labor required to calculate Ak’ for all pairs of systems from a given set. 37
CHEMICAL PHYSICS LETl-ERS
97. number 1
Volume
Many years ago, Casimir and Polder [Z] introduced a now widely-used identity for the ener,T denominators of eq. (Zb):
so that the matrix as
:
n,,, = -
k
i
‘z)
-
theorems
%”
=_-
+ %,
1 2ni I
c+i-
c-i-
dz (OOn - =)(.Q20,v + =) ’
qs2
~,(--p)IN)(JtI
exp(-iq
-D)
~,(~)lo,
w q.j(-q)IOA
(6)
(3
=
F
=
TZJ exp(iq
ii
exp(iq -$
(2=)3’4 VB(q) Substitution
of eqs. (5) and (6) in (1) yields c+i-
tic21
-RJ)_
= ire2 J
dz
jep-z exp(-ip-D)
c-ix J-d4
z
exp(-iq
f2
-D)
(01 ~,(P)h)(J4
n
WOIl
y&)10) -z
(7) Note that in eq. (7) the term with N = JZ = 0 is included whereas it was explicitly excluded in eq. (1). This is possible because the term gives rise to a double pole with zero residue and hence its contribution to the contour integraI is zero. Eq. (7) can be written more compactly with the help of the response functions
potential and
-rj+Di
=$Jdq
- D>sdq
and
X
with c < - QoN_ Neht , following Bethe [4] _ the Coulomb is represented by the Fourier integral IiiR,
x
~A(p)~JZ)(oI
C4) (27r)3/4 VA(q)
The convolution form of the separation theorem can be derived as follows. First the contour in eq. (4) is deformed so that it begins at c - i- where c < --R,. The contour goes to c + im and is then complcicd b) a large semi-cicle in the left half of the complex plane. The contribution from the semi-circle goes to zero as its radius goes to infinity_ Thus 1
in eq. (1) can be expressed
where dz (Won - Z)(q)N
The symbol above the integral sign in eq. (4) indicates that the contour encloses whl in the positive sense but does not enclose the pole at -ROX_ The separation rheorem based on eq. (4) will be referred to as the consolution form. It does not appear to be widely recognized that the two integral representations lead to two different separation theorems_ The purpose of this note is to show that the two are complementary in the sense that the Casimi-Polder form yields the dispersion energy whereas the convolution foml yields the total second-order interaction energy.
2. Separation
$Jc.lp pm2exp(-ip
x (01
The separation theorem based on eq. (3) will be referred IO as the Casimir-Polder form. Another integral representation for the energy denominators [3 ] is
Won
elements
Itool Wl?LWl’ =
(“&I)
6 May 1983
exp[-iq*(RJ
- ri +D)]qm2,
B,@,w)=~
iv
(01 q&J)INxNI ~,(q)lO) R ON- +z
WI
CHEMICAL
Volume 97, number 1
6 h&y 1983
PHYSICS LETTERS
Thus the convolution form of the separation theorem for the total second-order interaction energy can be written as &?)
= ins2 c-G=
xs
c-i-
FLJ
dp dq exp I-*-
@ + 411~-~4-~
(9)
x LQUz,~)~:GLW).
Additional equivalent expressions can be obtained by replacingA __@,q?)B,*(q,p,z) by B_@,q,zW&,p,z) or the symmetrized form
in which z&j and @ are the nuclear and electronic parts of VA and the states In) are now pure electronic states of a; the response functions of b are treated analogously_ Eq. (9) can now be written as MC21 = inm2
c+im X J c-i-
A derivation similar in spirit to the above but starting with eq_ (2b) and using the identity (3) yields the Casirnir-Polder separation theorem for the dispersion energy:
&sJQ
a4 exp[--iD*@+q)]p-2q-2
+A’d’~,q,z)B,@‘*(q,P,z) +A(d’(p,q.s)~~)*(q,~,z)l -
&$-$ = (2n2i)-' X J+
&JJdp
where the
dq exp[--iD-
>
(13
fast two terms are the induction terms and
the last term is the dispersion term. Similarly the response functions A and B* In the Casimir-Polder formula (10) can be replaced by Acd) and Bed)* respectively
(P + 4)1F24-2
where A (a) = A$i) + A 5) and B(d) = B,‘d’ + BF)_
x awzrZ)~*(qrPJ),
(10)
where A(p,q,z) = A_@,q,z) + A+(p,q,z) and similarly for B. Eq. (9) can be put in a more familiar form by observing that the states In> and IN) are combined nuclear -electronic states. When the nuclear coordinates are frozen during the determination of the intermolecular potentials, only the ground nuclear states can produce non-vanishing matrix elements of the nuclear coordinates and these will be ordinary (c-) numbers. Thus the response functions of eq. (8) can be decomposed into two terms: A,&,q,s)=A,@)@,q,z)
+--$d)@,q>z),
where Asp’(p,q,z) x [v?)(q)
= [v:‘(p)
+ (0 l~~%p)lO>]
i- (0 1l&q)1 O)] I(2 z),
(114
3. Concluding remarks Clearly the convolution form or the separation theorem is not equivalent to the Casimir-Polder form. On the contrary the two are complementary since the convolution form can be used to obtain the total secondorder interaction energy which consists of the induction and dispersion energies, and the Casimir-Polder form can be used to obtain the dispersion energy separately_ It is well known [S] that if the spectral forms (8) are not convenient, then the response functions can be obtained by the variational solution of first-order perturbation equations which describe the interactions of the separate systems with an electromagnetic plane wave. If desired the perturbation IVcan be expanded in multipoles to obtain separation theorems for the van der Waals coefficients in terms of dynamic multipole polarizabilities When the multipole expansion is employed for interactions between pairs of neutral closed-shell 39
Volume 97. number 1
CHEMICAL PHYSICS LETTERS
atoms the two separation theorems yield identical results because in such cases the induction energy is solely
and in part by the United States National Health Grant No. GM23223.
made up of exponentially decaying terms whose asymptotic expansion is identically zero. Finally we note that the present work can be generalized to exchange pertur-
References
bation theories and perturbative calculations of correlation energies in the manner demonstrated by Langhoff 151 for the Casimir-Polder separation theorem.
Acknowledgement This work was supported in part by the Natural and Engineering Research Council of Canada
Scicnccs
40
6 May 1983
Institutes
of
[ l] J-0. Hirschfelder and W.J. Meath. Advan. Chem. Phys. 12 (1967) 36. [2] H.B.G. Casimir and D. Polder. Phys. Rev. 73 (1948) 360. [ 31 R. Fulton, unpublished; B.J. Lauren& J. Chem. Phys. 50 (1969) 3576: Chem. Phys. Letters 5 (1970) 3: J. Chem. Phys. 56 (1972) 619: R. Boehm and R. Yaris. J. Chem. Phys. 55 (1971) 2620. [4] H.A. Bethe, Ann. Physik 5 (1930) 325. [ 5 ] P-W. Langhoff, Chem. Phys. Letters 20 (1973) 33.