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The dispersion interaction between closed-shell atoms. Accurate analytical expressions for multipole dispersion coefficients for the interaction between hydrogen atoms
CHEhlICAL
Volume 95, number 3
PHYSICS
4 March 1983
LETTERS
THE DlSPERSlON INTERACTION BETWEEN CLOSED-SHELL ATOMS. ACCURATE ANALYTICAL FOR THE INTERACTiON
EXPRESSIONS FOR MULTIPOLE DISPERSION COEFFICiENTS BETWEEN HYDROGEN ATOMS
R. LUYCKX * Fakulteit rau de Wetmschappen.
Vn#
Uuiversiteit Enmel.
B-l 050 Brussels. Belgium
Received 16 November 1962
A simple variation method is employed to obtain an expression for the dispersion interaction of closed-shell atoms in term of sum rule properties of the dynamic multipole polarizabilities of the interacting atoms. An analytical es&e&on for the dispersion cocfficicnts for the interaction of hydrogen atoms is obtained, using exact expressions for the sum NW of atomic hydropen.
1. Introduction The loag-range interaction between closed-shell atoms has been subject of thorough theoretical and experiment-d1investigation_Zt is well known that the dispersion interaction between closed-shell atoms can be represented by the series expansion [I]
#y(R) = -
c
tt=3
C?,,fR7” -
(1)
-
The van der Waals coefficients
C,,z can be written
as
WhCX CAB(lI, $) represents the contribution to the dispersion force, due to the interaction ofthe atom A with the 2’2~pole on atom B. It is the purpose of this letter to obtain analytical expressions cients CHw(Z1, 12) with an accuracy of a few promille.
2’1-pole on for the coeffi-
2. Theory
The coefficients
CAB(Z1, 1,) for the multipole
interaction
* Aangcsteld Navurscr Natiuuaal I’onds vour Wetenschappelijk
0 03 I-9163/S3/0000-0000/S
between
Ondcrzoek
03.00 0 1983 North-Holland
atoms A and B can be represented
as
(Bel_eium).
241
Volume 95. number
4 Msrch 1983
CHEMICAL PHYSICS LETTERS
3
where C$ (2‘) is the 2’r -dynamic multipole polarizabihty at imaginary frequencies of the atom A [2). Eq. (3) mak es clear that the problem of calculating the van der Waals coefticients can be effectively to the calculation of dynamic multipole polarizabilities. The dynamic ‘I-pole polarizability can be obtained evaluating [3,4] q(w)
= ($0 1Gj.0I x&J> * xl--‘4
where Q o is the multipole
I+,,) is the ground-state turbation equation
reduced
(4)
f
operator,
wave function
of the atom under consideration,
and G(o)
are the solutions
of the per-
(Ht,-Eo’O)I,~~(Wj)--~i~O)=O,
(5)
where H’1 is -eiven by ,v Hi = C'iY,*(,(f?;) i A solution
W
-
for x;(w)
can be obrained in the compiete
orthogonal
set of eigenstates 1$rr) and this yields
(6) (the primed summation polarhability
does not include
o[(oj
- G),
= C ‘1;: &E;l,i II -
whcrcff,~C, is called the oscillator
r,f.J, = lWc2~+
ljIEo.,,(~,,iH:IJlt,)
the ground state 1I$~}), Substituting
(6) in (4) we obtain
for the dynamic
(7) strength
and is given by
-
(3
Here IJ’n.,, is the transition frequency for a transition from the ground state 1I),-,>to the eigenstate j tin)_ The “exact” perturbation solution (7) for the dynamic polar&ability, satisfies sum rules for oscillator strengths and cscitation frequencies.
For k = 0 and I = I expression (8) reduces to the well-known Thomas-Reiche-Kuhn [5] sum rule c;rf$,rk: = N. where_W simply is the number of electrons of the atom. It is obvious that expression (7) for the dynamic polrtriznbilit ies is useless in actual calculations. However it is possible to construct accurate finite term approximations to the dynamic polarizabilities which satisfy a finite number of important sum rules, adopting a well chosen type of variation function. In fact the functional P [G(U)]
242
= <~r+(w)lHo -Eu
f wlxf(w))
- ($0 ]Hii$(w))
- (xlf(U)]H{]
+u)
(9
Volume95, number 3
CHEMICAL PHYSICS LETTERS
-4 March 1983
is the variational analog of the perturbation equation (5) [6]. Using as variation function
where the operators JIB will be defined below and miniiizing the functional L [cp(‘)‘(a)] with respect to the variation parameters cr;* (for the case of simplicity all superscripts I will be omitted tihenever no ambiguity is possible) one obtains c;(w)
= (N-’ NI+ o!);,;
,
(11)
where the matrices N and M are defined as [6] Mps = (~~I~~(~~
- &))A@ I Ift(-$+ C&JlA,W()
- qdAp
I Jt$ *
-N& = ~~fi&$q
I 9()> -
(12)
Using elementary matrix theory [7] expression (11) for the variation coefficients C;(W) can be expressed in its spectral form (13) where Afr = I, 2,.._, II are the eigenvalues of the matrix N-1 &‘Iand Rlf is the residue matrix corresponding
eigenvalue of A& Using (13) we obtain, as finite term appro~mation
to the
to the polarizability (14)
where the strength factors Zf are given by [6] .Zf”= [47r@Z -I- l)] Aj.&NIRjr] I1 _
(1%
It is possible to chose the operators $I) occurring in the variation function XI(O) in such a way that a limited number of sum rules will be satisfied exactly. In fact Including the perturbation Hi’) in the set of operators Ai automatically ensures the oscillator strength sum rulesSI(0) and the sum rule SI(-I) set of operators A$) = fli!
1,
where
Ff)
to be satisfied [8]_ Using as
= I?: ,
(16)
and f’d”l is defined recursively by [A,,~~lIrfi,~=~~_Il~,~.
(17)
closing on fg] I&>&J
I$,> =@I
9()> =H$
Go,,
the chose of If2 operator AI’), A$),
___,A$ ensures 2m sum rules s’(O), Sz(-I), ___~ S’(2m - l), to be satisfied. A little operator algebra shows that the matrix elementsMptq and Np,s can be expressed as [S]
N p,4 =S(2
- 4 - p)(2Z + l)f47r,
A$& = S(1 -p
- 4)(2Z + l)f4rr -
08)
An equivalent result has been obtained by Ianghoff [9]_ Thus using a variation function of the form (lo), together with the unknown operators (16), we obtain a matrix expression for the dynamic polarIzabiIity. The matrix elements are the sum rules S&k). These sum rules can be evaluated by quantum-mechanical calculations hydrogen atom analytical expressions for multipole sum rules were presented by Bell [lo] _
and for the
243
CHEMICAL
95. number 3
Volume
4
PHYSICS LETTERS
March 1983
TabIe 1
C 191 G ?_ c2.2 Cl,3
o-1 = (NG~M~,Q I,,, + I, 0 NI,~Q~) This can be done analytically
One-parameter (24)
Two-parameter (29)
Accurate
6.0 57.9 1063 1008
6.48 62.0 1133 1073
6.499 62.2 1135 1075
I131
(75)
NI,lM lb_ Using a two-parameter
using the special form of the matrix
variation
func-
tion we have 1~1= m’ = 2 and
where a1 =
Sz(O) N-3
$(-1)
- Sz(O) Sz(-3)
a2 - rq(-!)
Sz(--3) - $(__2)
- Sr(-1) Sz(-2)
Sz(-l)Sz(--3)
-S&--2)
’
an anaIog form for N;,‘M,,_ The dispersion _._,S1(--3) as
with
For 1, = lb the expression (4[,)! C&I,
S,(-2)
=
(2
Substituting
-I-
coefficient
S&-l)
Sza(-3)
-
S&l)
2$(-l) &J-3)
S&3)
and S,(-3)
2)!(2 -i- 3)
= (21)!(~4~-+;~]].;~2;
the analytic
can now be expressed
in term of the SW~ rulesSz(0),
SI(q
+SzJO)
s&u
* S$-2)
the sum rules 51(-Z)
qr + 1)221*3
CW
’
becomes very simple and reduces to
= S(21,)! -
For hydrogen
-_
can be calculated
and are given by
101-k 2) _
(33)
’
expressions
for the sum rules $(O), ___> SI(--3) of hydrogen,
in (3s) one obtains
245
&I”
(2z3 + 2&)! (la.lb) = 22 (Ia+lb )
(2Z,’+ 4za +
(
+------+ (2,
4 March 1983
CHEMICAL PHYSICS LETTERS
Volume 95. number 3
I)
+ 3)(la + 1)
21; + 41, + 1 -[#a
+ 1)
(2; + 4&
+
21; + 4Ib + 1 l&b + 1) (” + lxzb + ‘)
+ 1)
(a,,
+ I)(&
@b + 3)(zb -I- l)
+ 2)(2z, + ‘,(‘a +
2,
9(la -I,,)' +zl;llb(g, +3)(& IL 3(1, - I&(21,
2zaib
+ zz, + 5) +3)
&,(&,
+ 3)(&
+ 3,
(29) In table 1 we present the dispersion coefficients for the long-range interaction of hydrogen atoms, calculated using the analytic espression (29). They are accurate to within 0.3% for the dipole-dipole interaction and even better for higher-order mulripole dispersion coefficients_ Moreover it can be proven that using the sum rules S@), --- , Sl(-3) one obtains a lower bound to the exact dispersion coefficient [ 151. Acknowledgement 1 would like to thank the “Nationaal
Fonds voor Wetenschappelijk
Onderzoek
(Belgium)”
for financial
References [ I] A. Datgnrno and W.D. Davison, .Advan. AI Mol. Phys. 2 (1966) 1. 12 1 tLl3.G. Casimir and D. Poldrr, Phys. Rev. 73 (1948) 360. 131 A. Dalgarno. Advan. Chem. Phys. 12 (1966) 143_ 14 ] K. Luycks. Ph. Coulon and H.N.W_ Lekkerkerker. J. Chem. Phys. 69 (1978) 2424. [SJ IV. Kuhn, Z. I’hysik 33 (1925) 408. I=. Reicbx and W. Thomas. 2. Physik 34 (192 5) 5 10. 161 R. Luycks. Theory rend Calcuiarion of Dispersion Forces between Closed Shell Atoms. Doctoral R. Luycks. Ph. Couion and H.N.W. Lekkerkerkcr, Chem. Phys. Letters 48 (1977) 187. [ 7 ] I’. Lancaster. Theory of matrices (Academic Press, New York. 1969). [S] K. Luycks. Chcm. Phys. Letters 9.5 (1983) 235. 19 ] I’.\\‘. Lxnghoff. J. Sims and CT. Corcoran, Phys. Rev. A10 (1974) 829. [lo] R.G. Bell. Proc. Phys. Sot. (London) 92 (1967) 842. [ I1 ] R. Luycks. Ph. Coulon and H.N.W. Lekkerkerker. J. Chcm. Phys. 71 (1979) 4734. [ 12 J L. Pauling and J.Y. Beach. Phys. Rev. 47 (1935) 686. 113] Y.hl. Chan and A. Dnlgarno, hlol. Phys. 9 (1969) 1349. 1141 R. ~uycks. I. Dclbacn. Ph. Coulon and H.N.W. Lekkerkerker. Phys. Rev. A19 (1979) 324.
Thesis;
support.
Volume 95, number 3
PHYSICS
4 March 1983
LETTERS
THE DlSPERSlON INTERACTION BETWEEN CLOSED-SHELL ATOMS. ACCURATE ANALYTICAL FOR THE INTERACTiON
EXPRESSIONS FOR MULTIPOLE DISPERSION COEFFICiENTS BETWEEN HYDROGEN ATOMS
R. LUYCKX * Fakulteit rau de Wetmschappen.
Vn#
Uuiversiteit Enmel.
B-l 050 Brussels. Belgium
Received 16 November 1962
A simple variation method is employed to obtain an expression for the dispersion interaction of closed-shell atoms in term of sum rule properties of the dynamic multipole polarizabilities of the interacting atoms. An analytical es&e&on for the dispersion cocfficicnts for the interaction of hydrogen atoms is obtained, using exact expressions for the sum NW of atomic hydropen.
1. Introduction The loag-range interaction between closed-shell atoms has been subject of thorough theoretical and experiment-d1investigation_Zt is well known that the dispersion interaction between closed-shell atoms can be represented by the series expansion [I]
#y(R) = -
c
tt=3
C?,,fR7” -
(1)
-
The van der Waals coefficients
C,,z can be written
as
WhCX CAB(lI, $) represents the contribution to the dispersion force, due to the interaction ofthe atom A with the 2’2~pole on atom B. It is the purpose of this letter to obtain analytical expressions cients CHw(Z1, 12) with an accuracy of a few promille.
2’1-pole on for the coeffi-
2. Theory
The coefficients
CAB(Z1, 1,) for the multipole
interaction
* Aangcsteld Navurscr Natiuuaal I’onds vour Wetenschappelijk
0 03 I-9163/S3/0000-0000/S
between
Ondcrzoek
03.00 0 1983 North-Holland
atoms A and B can be represented
as
(Bel_eium).
241
Volume 95. number
4 Msrch 1983
CHEMICAL PHYSICS LETTERS
3
where C$ (2‘) is the 2’r -dynamic multipole polarizabihty at imaginary frequencies of the atom A [2). Eq. (3) mak es clear that the problem of calculating the van der Waals coefticients can be effectively to the calculation of dynamic multipole polarizabilities. The dynamic ‘I-pole polarizability can be obtained evaluating [3,4] q(w)
= ($0 1Gj.0I x&J> * xl--‘4
where Q o is the multipole
I+,,) is the ground-state turbation equation
reduced
(4)
f
operator,
wave function
of the atom under consideration,
and G(o)
are the solutions
of the per-
(Ht,-Eo’O)I,~~(Wj)--~i~O)=O,
(5)
where H’1 is -eiven by ,v Hi = C'iY,*(,(f?;) i A solution
W
-
for x;(w)
can be obrained in the compiete
orthogonal
set of eigenstates 1$rr) and this yields
(6) (the primed summation polarhability
does not include
o[(oj
- G),
= C ‘1;: &E;l,i II -
whcrcff,~C, is called the oscillator
r,f.J, = lWc2~+
ljIEo.,,(~,,iH:IJlt,)
the ground state 1I$~}), Substituting
(6) in (4) we obtain
for the dynamic
(7) strength
and is given by
-
(3
Here IJ’n.,, is the transition frequency for a transition from the ground state 1I),-,>to the eigenstate j tin)_ The “exact” perturbation solution (7) for the dynamic polar&ability, satisfies sum rules for oscillator strengths and cscitation frequencies.
For k = 0 and I = I expression (8) reduces to the well-known Thomas-Reiche-Kuhn [5] sum rule c;rf$,rk: = N. where_W simply is the number of electrons of the atom. It is obvious that expression (7) for the dynamic polrtriznbilit ies is useless in actual calculations. However it is possible to construct accurate finite term approximations to the dynamic polarizabilities which satisfy a finite number of important sum rules, adopting a well chosen type of variation function. In fact the functional P [G(U)]
242
= <~r+(w)lHo -Eu
f wlxf(w))
- ($0 ]Hii$(w))
- (xlf(U)]H{]
+u)
(9
Volume95, number 3
CHEMICAL PHYSICS LETTERS
-4 March 1983
is the variational analog of the perturbation equation (5) [6]. Using as variation function
where the operators JIB will be defined below and miniiizing the functional L [cp(‘)‘(a)] with respect to the variation parameters cr;* (for the case of simplicity all superscripts I will be omitted tihenever no ambiguity is possible) one obtains c;(w)
= (N-’ NI+ o!);,;
,
(11)
where the matrices N and M are defined as [6] Mps = (~~I~~(~~
- &))A@ I Ift(-$+ C&JlA,W()
- qdAp
I Jt$ *
-N& = ~~fi&$q
I 9()> -
(12)
Using elementary matrix theory [7] expression (11) for the variation coefficients C;(W) can be expressed in its spectral form (13) where Afr = I, 2,.._, II are the eigenvalues of the matrix N-1 &‘Iand Rlf is the residue matrix corresponding
eigenvalue of A& Using (13) we obtain, as finite term appro~mation
to the
to the polarizability (14)
where the strength factors Zf are given by [6] .Zf”= [47r@Z -I- l)] Aj.&NIRjr] I1 _
(1%
It is possible to chose the operators $I) occurring in the variation function XI(O) in such a way that a limited number of sum rules will be satisfied exactly. In fact Including the perturbation Hi’) in the set of operators Ai automatically ensures the oscillator strength sum rulesSI(0) and the sum rule SI(-I) set of operators A$) = fli!
1,
where
Ff)
to be satisfied [8]_ Using as
= I?: ,
(16)
and f’d”l is defined recursively by [A,,~~lIrfi,~=~~_Il~,~.
(17)
closing on fg] I&>&J
I$,> =@I
9()> =H$
Go,,
the chose of If2 operator AI’), A$),
___,A$ ensures 2m sum rules s’(O), Sz(-I), ___~ S’(2m - l), to be satisfied. A little operator algebra shows that the matrix elementsMptq and Np,s can be expressed as [S]
N p,4 =S(2
- 4 - p)(2Z + l)f47r,
A$& = S(1 -p
- 4)(2Z + l)f4rr -
08)
An equivalent result has been obtained by Ianghoff [9]_ Thus using a variation function of the form (lo), together with the unknown operators (16), we obtain a matrix expression for the dynamic polarIzabiIity. The matrix elements are the sum rules S&k). These sum rules can be evaluated by quantum-mechanical calculations hydrogen atom analytical expressions for multipole sum rules were presented by Bell [lo] _
and for the
243
CHEMICAL
95. number 3
Volume
4
PHYSICS LETTERS
March 1983
TabIe 1
C 191 G ?_ c2.2 Cl,3
o-1 = (NG~M~,Q I,,, + I, 0 NI,~Q~) This can be done analytically
One-parameter (24)
Two-parameter (29)
Accurate
6.0 57.9 1063 1008
6.48 62.0 1133 1073
6.499 62.2 1135 1075
I131
(75)
NI,lM lb_ Using a two-parameter
using the special form of the matrix
variation
func-
tion we have 1~1= m’ = 2 and
where a1 =
Sz(O) N-3
$(-1)
- Sz(O) Sz(-3)
a2 - rq(-!)
Sz(--3) - $(__2)
- Sr(-1) Sz(-2)
Sz(-l)Sz(--3)
-S&--2)
’
an anaIog form for N;,‘M,,_ The dispersion _._,S1(--3) as
with
For 1, = lb the expression (4[,)! C&I,
S,(-2)
=
(2
Substituting
-I-
coefficient
S&-l)
Sza(-3)
-
S&l)
2$(-l) &J-3)
S&3)
and S,(-3)
2)!(2 -i- 3)
= (21)!(~4~-+;~]].;~2;
the analytic
can now be expressed
in term of the SW~ rulesSz(0),
SI(q
+SzJO)
s&u
* S$-2)
the sum rules 51(-Z)
qr + 1)221*3
CW
’
becomes very simple and reduces to
= S(21,)! -
For hydrogen
-_
can be calculated
and are given by
101-k 2) _
(33)
’
expressions
for the sum rules $(O), ___> SI(--3) of hydrogen,
in (3s) one obtains
245
&I”
(2z3 + 2&)! (la.lb) = 22 (Ia+lb )
(2Z,’+ 4za +
(
+------+ (2,
4 March 1983
CHEMICAL PHYSICS LETTERS
Volume 95. number 3
I)
+ 3)(la + 1)
21; + 41, + 1 -[#a
+ 1)
(2; + 4&
+
21; + 4Ib + 1 l&b + 1) (” + lxzb + ‘)
+ 1)
(a,,
+ I)(&
@b + 3)(zb -I- l)
+ 2)(2z, + ‘,(‘a +
2,
9(la -I,,)' +zl;llb(g, +3)(& IL 3(1, - I&(21,
2zaib
+ zz, + 5) +3)
&,(&,
+ 3)(&
+ 3,
(29) In table 1 we present the dispersion coefficients for the long-range interaction of hydrogen atoms, calculated using the analytic espression (29). They are accurate to within 0.3% for the dipole-dipole interaction and even better for higher-order mulripole dispersion coefficients_ Moreover it can be proven that using the sum rules S@), --- , Sl(-3) one obtains a lower bound to the exact dispersion coefficient [ 151. Acknowledgement 1 would like to thank the “Nationaal
Fonds voor Wetenschappelijk
Onderzoek
(Belgium)”
for financial
References [ I] A. Datgnrno and W.D. Davison, .Advan. AI Mol. Phys. 2 (1966) 1. 12 1 tLl3.G. Casimir and D. Poldrr, Phys. Rev. 73 (1948) 360. 131 A. Dalgarno. Advan. Chem. Phys. 12 (1966) 143_ 14 ] K. Luycks. Ph. Coulon and H.N.W_ Lekkerkerker. J. Chem. Phys. 69 (1978) 2424. [SJ IV. Kuhn, Z. I’hysik 33 (1925) 408. I=. Reicbx and W. Thomas. 2. Physik 34 (192 5) 5 10. 161 R. Luycks. Theory rend Calcuiarion of Dispersion Forces between Closed Shell Atoms. Doctoral R. Luycks. Ph. Couion and H.N.W. Lekkerkerkcr, Chem. Phys. Letters 48 (1977) 187. [ 7 ] I’. Lancaster. Theory of matrices (Academic Press, New York. 1969). [S] K. Luycks. Chcm. Phys. Letters 9.5 (1983) 235. 19 ] I’.\\‘. Lxnghoff. J. Sims and CT. Corcoran, Phys. Rev. A10 (1974) 829. [lo] R.G. Bell. Proc. Phys. Sot. (London) 92 (1967) 842. [ I1 ] R. Luycks. Ph. Coulon and H.N.W. Lekkerkerker. J. Chcm. Phys. 71 (1979) 4734. [ 12 J L. Pauling and J.Y. Beach. Phys. Rev. 47 (1935) 686. 113] Y.hl. Chan and A. Dnlgarno, hlol. Phys. 9 (1969) 1349. 1141 R. ~uycks. I. Dclbacn. Ph. Coulon and H.N.W. Lekkerkerker. Phys. Rev. A19 (1979) 324.
Thesis;
support.