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Van Der Waals force constants from non-integer oscillator strength sums
Volume 5. number .3
VAN
CHEMICAL
DER
WAALS
FORCE
PHYSICS LETTERS
CONSTANTS
OSCILLATOR
L5 QriI
FROM
STRENGTH
1970
NON-INTEGER
SUMS *
RUSSELL T. PACK Depadmwt
of Cl~rn~islYJ, Brigham
Yoan_.q10zizxvsi1.y. Provo. Unlr 841‘01. LSA
Received 16 January 1970
A simple formula involving the oscillator strength sum S(-3/2) gives an upper boundfor the Van der Waals constant Cab. Modifications of the formula give Cab for all pairs ab of the atoms R, He, Se. r\r. Kr, and Se within an error of 0.3%to 12.1%.
The oscillator
strength sums,
s(k),
are
given
over
k of the S(k) if the S(K) are known for szv-
era1 integer values of k.
by
S(h)
=c’f011Ek’ n
(1)
R
An even simpler procedure is as follows. (3) can be rewritten in the form,
Eq.
where f on is the average dipole oscillator strength,
and the E,z = E, - Eg are atomic excitation energies **. These S(k) are known to be useful in the calculation of many atomic properties (see, e.g.,
ref. Cl]). However, only the s(k) for integer values of k seem to have found much use. Our purpose here is to show that a simple relation exists between the van der Waals force constant and S( -3,‘2). The magnetic quantum number-averaged van der Waals constant in the long-range interatomic potential, r’ab = -Cab/R6, for the interaction of atoms a and b. is given by
A ,&a,
b) is the ratio
and is always fore,
less
which Cab can then be obtained
f h from
is now well
known (see, e.g., ref. [2]): Bell [S] has shown that Cab can be expreMed. accurately as a sum
* Research supported in part by a grant from the Brigham Young University Research Fund. l * All quantities herein are in hartree atomic units. Most of the references units.
cited USC Rydberg ener,gv
mean of
mean
than or equal to unity; there-
one has immediately
that
Cab s (3,‘4)s,( - 3/‘2)sb( - 3/‘2) This simple formula is itself a rather per bound for the interaction constant atoms (see table 1). For unlike atoms age energy approximation reduces eq. improved result, Cab = (3/4@(a,
The use of the S(k) to calculate
of the geometric
the excitation energies to their arithmetic
Because A&a,
(6) good upfor Like an aver(4) to an
b)Sa(- 3/2)Sb( - 3/2) -
(7)
b) is usually near unity, this
approximation is much less drastic than the average
energy
approximations
required
to get
C at, in terms of S(- 2) (the London formula) or
S( - 1) (UnsMd’s approximation).
We have caku-
lated Cab from (7) for all pair& of hydrogen and noble gas atoms; the results are in table 2. The.diiect calculation of S(- 3/Z} is expected to be difficult. The S( - 3/2) used here were obtained by interpolation using the form suggested 257
Volume 5, number 5
CHEMICAL PHYSICS LETTERS
15 April 1970
TabLe I
Atomic Quantities.
The properties
tained from a many-term
listed are, respectively,
non-refativistic
WerpoIation
formtda,values of S(-3/2) from a many-term
the first excitation energies,
formula,
nrefativistic”
formula, the upper bound on Caa obtained from eq. (6)
and S(n$ 1) (- 3/z), and the aL,curate values of Caa of ref. [S]. All quantities H
He
the values of S(- 3/Z) ob-
values of S(- 3/z) from a two-term non-relativistic
Ne
are in hartree
AZ
atomic units
Kr
Xe
El
0.37500
O.F972
0.61916
0.43467
(-3/q
2.97636
1.4!581
3.0227
9.6070
13.859
20.135
S(m2) (-3/2)
2.97061
1.41645
2.9752
9.0159
12.532
17.507
Sfr.')(-3/2)
2.97020
1.41590
3.ol.02
6.617 6.500
1.5Q3
6.853
69.2
144.1
304.1
1.456
6.305
65.3
130-a
268.2
&if)
Caa (upper bound) %a (ref. 1511
0.39115
9.5146
13.688
0.35168
19.605
Table 2
Van der Waals Coefficients Cab. Those labeled by (I)are thenon-relativistic resultsof the presentwork;thoseIa-
bcfed by (2) are the resulta
of the present work using “relativistic n formulas; f5f. All are in hartreeatomicunits
H H (1)
He
Ar
Ne
and those Iabeled by (3) are from ref.
Kr
Xe
(2) (3)
6.521 6.390 6.500
2.921 2.873 2.815
6.44 6.30 5.62
21.04 20.42 19.93
30.42 29.44 28.54
44.18 42.33 41.61
Ho (1) (2) (3)
2.921 2.873 2.815
1:4a2 1.452 1.456
3.143 3.069 3.012
9.66 9.40 9.62
13.72 13.33 13.44
19.58 18.84 18.67
Ne (1) (2)
6.30
3.069
30.20
43.33 41.44
. 131
5.62
9.W
(1)
21.04
9.66
21.15
::;
20.42 19.93
9.40 9.62
20.46 19.60
Iir(I) (2) (3)
30.42 29.44 28.54
13:72 13.33 x3.44
Xo 11) (2) (3)
44.18 42.33 42.61
19.58 18.84 18.67
Ar
by Dalgarna and Kingston
6.44
3.143
6.753
6.563
21.1s
20.46
29.16
19.60
27.26
68.2
98.3
142.2
65.3 65.6
92.1 94.2
135.0 130.4
30.26 29.16 %27.26
98.3 94.2 92.1
142.0 136.7 130.4
206.0 194.9 186.7
43.33 41.44 37.50
142.2 135.0 130.4
206.0 194.9 186.7
299.6 280.7 268.2
6.905
31.50
s(=, 1) (-3/Z), were obtained with K= I, consistent with the divergence ['?I of the relativistic S(1). In order to see how many dais axe neces-
[4],
S(k) =
sary to obtain accurate values of 5(3/Z), the S(- 3/2) were afao calculatedusing an equation of form (‘if containing only two constants chosen to
The ai were chosen to fit the values of S(k), k= - 2 to - 6, given by Bell and Kingston [5]. For H and He the formula was also made to fit accuratevalues 14, S] of S(-I). Two vdiues of K were used. Non-relativistic S(- 3/2), the S(n,l) (- 3/2) of table 1, were obtained using K = 5/2, consis-
tent with the known divergence of the non-reiativistic 5(5/Z). *sa
The “relativistic”
estimates,
fit S(- 2) and the value of S(- 1) determined by the many term formula used above. The results are
the S(nt 2)( - 3/Z) in table 1. The S(‘g 2)( - 3/2) were very similar to the S(% 2)(- 3/2) and were amitted from the table. It appears that S(- 1) and S(-2) are sufficient to accurately determine S(- 3j2) for small atoms but not for large atoms. To calculate reasonable values for Afa, b) the sumsin (4) w&e replacedby integralsandaor-
Volume 5, number 5
CHEMICALPHYSICSLETTERS
15 April 1970
cab simply represent
malized,
differences
in the ability
of our simple formulas to fit the data smoothly.
(9)
where El (see table 1) is the energy of the lowest excited state for whichfO1 is non zero. For the non-relativistic estimate, f(x) was then replaced by xe7j2,
th e simplest
distribution
consistent
with the convergence of S(2) and divergence of
S(5/2),
ih)(a,
This yields
b) = 8
+r-Btan-$“)
[r8t~-i(r-1) -r-7+,-5/3
-
+r5/3- Y3/5+r/7
-Y-3/5+7=1/7]
,
The calculations of BelI and Kingston require no such formulas and should be accurate for atoms of any size. While the method used here is less accurate than Bell’s method if a large number of s(k) are known, it is more accurate than other semiempirical methods [9] (London, Slater-Kirkwood, etc.) of similar simplicity. Our procedure should be most useful in calculating C& ior pairs of fairly small atoms for which few S(k) are known. Reasonable values of Cab can be obtained from a knowledge of only ~l..S(Cl),S(-l1!,
and S(- 2) for the atoms involved. S(0) =Z is the
atomic number, and ~1 is Mown [LOI for aI1 atoms. S( - 2) = cz is the dipole polarizabilify and is known for most atoms, and
a- 1) = ~(OlC~r$2
(10)
(b)]1’2- For the atoms considered here, this A-! “)(a, b) varies from a high of 0.9854 for like atoms to a low of 0.9157 for the He-Xe interaction. The “relativistic ’ estimate was obtained by replacing f (x) by x-2 to account for the divergence of the relativistic S(1). The result is
where r =[el(a)/e
can be calculated from ground state atomic wave
functions. Finally, we note that the procedure just described is easily extended to the interaction of three atoms. The long range potential has the from [5] V
_@)(a, b) =;[2r51n(l + 4)
- 2,~’+ Y
+2r-51n(l+r2)-2r-3+V1].
(11)
dr)(a, b) varied from a high of 0.965’7 for like atoms to a low of 0.9014 for He-Xe. The values of cab calculated using the A(a, b) and S(l)(- 3/2) are listed in table 2. Also listed for comparison are the values of Bell and Kingston [5] determined by Bell’s [3] method and listed by Dalgarno [2] at the best available. (The values cf CHH is known much more accurately than this [8]. ) Our non-relativistic results agree will with those of Bell and Kingston for small atoms,
the difference
being only 0.3% for
CHN_
However, the differences are much larger for the larger atoms. Our “relativistic”
results are within 5% of Bell and Kingston’s results for all the cab except some of those involving Ne, in which the difference is as large as 12.1% It should be noted that the differences between the Cab calculated using the non-relativistic and “relativistic” formulas do not represent relativistic corrections. Relativistic corrections to the S(k) for all negative k are expected to be small due to the fact that most of the contributions to those sums come from transitions involving only
the valence electrons. The differences in the
(0)
abc
=c abc(l + 3 cos e1 cos oacos 83)/i?$+;
,
and the upper bound obtained for like atoms is C aa
s (g/16)Sat- 2) [Sat- 3/2)]’ -
!W
For the interaction of three Ii atoms this gives CHHH 4 22.3333 which is only 3.2% higher than
21.6425, the best known value [2]. REFERENCES [l] 6.O.Hirschfelder. Epstein,
Advan.
W.Byers Quantum
Broxn and S.T.
Chem.
I (1964)
228.
A. Dalgamo, Advan.Chem.Phys. 12 (1967) W3. [3] R. J. Be& Proc. Phys.Soc. (London) 86 (1966) 17_ [4] A.Dalgarno and kE. Kingston, PrOC.ROy. SOC. [Z]
A259 (1960) 424. [5] R. J. Bell and A. E. Kingston, Proc. Phys. SCC. (London) 88 (1966) 901. [S] C. L. Pekeris, Phys. Rev. 213 (1959) i216. [I’] K. R. Piech and J. S. Lewinger. Phys. Rev. 135A
(19G) 332. [8] M. N. Adamov, M. D. Balm&x and T.K. Rebate, Intern. J. QuantumChem. 3 (1969) 13. 191L. Salem. bfoi. Phys. 3 (1960) 441. [lo] C. E. Moore, Atomic energy levels. Circular 467, vol. 1 (1949). Vol. 2 (1958). Vol. 3 (1958) (Natl. Bur. Std., Washington).
259
VAN
CHEMICAL
DER
WAALS
FORCE
PHYSICS LETTERS
CONSTANTS
OSCILLATOR
L5 QriI
FROM
STRENGTH
1970
NON-INTEGER
SUMS *
RUSSELL T. PACK Depadmwt
of Cl~rn~islYJ, Brigham
Yoan_.q10zizxvsi1.y. Provo. Unlr 841‘01. LSA
Received 16 January 1970
A simple formula involving the oscillator strength sum S(-3/2) gives an upper boundfor the Van der Waals constant Cab. Modifications of the formula give Cab for all pairs ab of the atoms R, He, Se. r\r. Kr, and Se within an error of 0.3%to 12.1%.
The oscillator
strength sums,
s(k),
are
given
over
k of the S(k) if the S(K) are known for szv-
era1 integer values of k.
by
S(h)
=c’f011Ek’ n
(1)
R
An even simpler procedure is as follows. (3) can be rewritten in the form,
Eq.
where f on is the average dipole oscillator strength,
and the E,z = E, - Eg are atomic excitation energies **. These S(k) are known to be useful in the calculation of many atomic properties (see, e.g.,
ref. Cl]). However, only the s(k) for integer values of k seem to have found much use. Our purpose here is to show that a simple relation exists between the van der Waals force constant and S( -3,‘2). The magnetic quantum number-averaged van der Waals constant in the long-range interatomic potential, r’ab = -Cab/R6, for the interaction of atoms a and b. is given by
A ,&a,
b) is the ratio
and is always fore,
less
which Cab can then be obtained
f h from
is now well
known (see, e.g., ref. [2]): Bell [S] has shown that Cab can be expreMed. accurately as a sum
* Research supported in part by a grant from the Brigham Young University Research Fund. l * All quantities herein are in hartree atomic units. Most of the references units.
cited USC Rydberg ener,gv
mean of
mean
than or equal to unity; there-
one has immediately
that
Cab s (3,‘4)s,( - 3/‘2)sb( - 3/‘2) This simple formula is itself a rather per bound for the interaction constant atoms (see table 1). For unlike atoms age energy approximation reduces eq. improved result, Cab = (3/4@(a,
The use of the S(k) to calculate
of the geometric
the excitation energies to their arithmetic
Because A&a,
(6) good upfor Like an aver(4) to an
b)Sa(- 3/2)Sb( - 3/2) -
(7)
b) is usually near unity, this
approximation is much less drastic than the average
energy
approximations
required
to get
C at, in terms of S(- 2) (the London formula) or
S( - 1) (UnsMd’s approximation).
We have caku-
lated Cab from (7) for all pair& of hydrogen and noble gas atoms; the results are in table 2. The.diiect calculation of S(- 3/Z} is expected to be difficult. The S( - 3/2) used here were obtained by interpolation using the form suggested 257
Volume 5, number 5
CHEMICAL PHYSICS LETTERS
15 April 1970
TabLe I
Atomic Quantities.
The properties
tained from a many-term
listed are, respectively,
non-refativistic
WerpoIation
formtda,values of S(-3/2) from a many-term
the first excitation energies,
formula,
nrefativistic”
formula, the upper bound on Caa obtained from eq. (6)
and S(n$ 1) (- 3/z), and the aL,curate values of Caa of ref. [S]. All quantities H
He
the values of S(- 3/Z) ob-
values of S(- 3/z) from a two-term non-relativistic
Ne
are in hartree
AZ
atomic units
Kr
Xe
El
0.37500
O.F972
0.61916
0.43467
(-3/q
2.97636
1.4!581
3.0227
9.6070
13.859
20.135
S(m2) (-3/2)
2.97061
1.41645
2.9752
9.0159
12.532
17.507
Sfr.')(-3/2)
2.97020
1.41590
3.ol.02
6.617 6.500
1.5Q3
6.853
69.2
144.1
304.1
1.456
6.305
65.3
130-a
268.2
&if)
Caa (upper bound) %a (ref. 1511
0.39115
9.5146
13.688
0.35168
19.605
Table 2
Van der Waals Coefficients Cab. Those labeled by (I)are thenon-relativistic resultsof the presentwork;thoseIa-
bcfed by (2) are the resulta
of the present work using “relativistic n formulas; f5f. All are in hartreeatomicunits
H H (1)
He
Ar
Ne
and those Iabeled by (3) are from ref.
Kr
Xe
(2) (3)
6.521 6.390 6.500
2.921 2.873 2.815
6.44 6.30 5.62
21.04 20.42 19.93
30.42 29.44 28.54
44.18 42.33 41.61
Ho (1) (2) (3)
2.921 2.873 2.815
1:4a2 1.452 1.456
3.143 3.069 3.012
9.66 9.40 9.62
13.72 13.33 13.44
19.58 18.84 18.67
Ne (1) (2)
6.30
3.069
30.20
43.33 41.44
. 131
5.62
9.W
(1)
21.04
9.66
21.15
::;
20.42 19.93
9.40 9.62
20.46 19.60
Iir(I) (2) (3)
30.42 29.44 28.54
13:72 13.33 x3.44
Xo 11) (2) (3)
44.18 42.33 42.61
19.58 18.84 18.67
Ar
by Dalgarna and Kingston
6.44
3.143
6.753
6.563
21.1s
20.46
29.16
19.60
27.26
68.2
98.3
142.2
65.3 65.6
92.1 94.2
135.0 130.4
30.26 29.16 %27.26
98.3 94.2 92.1
142.0 136.7 130.4
206.0 194.9 186.7
43.33 41.44 37.50
142.2 135.0 130.4
206.0 194.9 186.7
299.6 280.7 268.2
6.905
31.50
s(=, 1) (-3/Z), were obtained with K= I, consistent with the divergence ['?I of the relativistic S(1). In order to see how many dais axe neces-
[4],
S(k) =
sary to obtain accurate values of 5(3/Z), the S(- 3/2) were afao calculatedusing an equation of form (‘if containing only two constants chosen to
The ai were chosen to fit the values of S(k), k= - 2 to - 6, given by Bell and Kingston [5]. For H and He the formula was also made to fit accuratevalues 14, S] of S(-I). Two vdiues of K were used. Non-relativistic S(- 3/2), the S(n,l) (- 3/2) of table 1, were obtained using K = 5/2, consis-
tent with the known divergence of the non-reiativistic 5(5/Z). *sa
The “relativistic”
estimates,
fit S(- 2) and the value of S(- 1) determined by the many term formula used above. The results are
the S(nt 2)( - 3/Z) in table 1. The S(‘g 2)( - 3/2) were very similar to the S(% 2)(- 3/2) and were amitted from the table. It appears that S(- 1) and S(-2) are sufficient to accurately determine S(- 3j2) for small atoms but not for large atoms. To calculate reasonable values for Afa, b) the sumsin (4) w&e replacedby integralsandaor-
Volume 5, number 5
CHEMICALPHYSICSLETTERS
15 April 1970
cab simply represent
malized,
differences
in the ability
of our simple formulas to fit the data smoothly.
(9)
where El (see table 1) is the energy of the lowest excited state for whichfO1 is non zero. For the non-relativistic estimate, f(x) was then replaced by xe7j2,
th e simplest
distribution
consistent
with the convergence of S(2) and divergence of
S(5/2),
ih)(a,
This yields
b) = 8
+r-Btan-$“)
[r8t~-i(r-1) -r-7+,-5/3
-
+r5/3- Y3/5+r/7
-Y-3/5+7=1/7]
,
The calculations of BelI and Kingston require no such formulas and should be accurate for atoms of any size. While the method used here is less accurate than Bell’s method if a large number of s(k) are known, it is more accurate than other semiempirical methods [9] (London, Slater-Kirkwood, etc.) of similar simplicity. Our procedure should be most useful in calculating C& ior pairs of fairly small atoms for which few S(k) are known. Reasonable values of Cab can be obtained from a knowledge of only ~l..S(Cl),S(-l1!,
and S(- 2) for the atoms involved. S(0) =Z is the
atomic number, and ~1 is Mown [LOI for aI1 atoms. S( - 2) = cz is the dipole polarizabilify and is known for most atoms, and
a- 1) = ~(OlC~r$2
(10)
(b)]1’2- For the atoms considered here, this A-! “)(a, b) varies from a high of 0.9854 for like atoms to a low of 0.9157 for the He-Xe interaction. The “relativistic ’ estimate was obtained by replacing f (x) by x-2 to account for the divergence of the relativistic S(1). The result is
where r =[el(a)/e
can be calculated from ground state atomic wave
functions. Finally, we note that the procedure just described is easily extended to the interaction of three atoms. The long range potential has the from [5] V
_@)(a, b) =;[2r51n(l + 4)
- 2,~’+ Y
+2r-51n(l+r2)-2r-3+V1].
(11)
dr)(a, b) varied from a high of 0.965’7 for like atoms to a low of 0.9014 for He-Xe. The values of cab calculated using the A(a, b) and S(l)(- 3/2) are listed in table 2. Also listed for comparison are the values of Bell and Kingston [5] determined by Bell’s [3] method and listed by Dalgarno [2] at the best available. (The values cf CHH is known much more accurately than this [8]. ) Our non-relativistic results agree will with those of Bell and Kingston for small atoms,
the difference
being only 0.3% for
CHN_
However, the differences are much larger for the larger atoms. Our “relativistic”
results are within 5% of Bell and Kingston’s results for all the cab except some of those involving Ne, in which the difference is as large as 12.1% It should be noted that the differences between the Cab calculated using the non-relativistic and “relativistic” formulas do not represent relativistic corrections. Relativistic corrections to the S(k) for all negative k are expected to be small due to the fact that most of the contributions to those sums come from transitions involving only
the valence electrons. The differences in the
(0)
abc
=c abc(l + 3 cos e1 cos oacos 83)/i?$+;
,
and the upper bound obtained for like atoms is C aa
s (g/16)Sat- 2) [Sat- 3/2)]’ -
!W
For the interaction of three Ii atoms this gives CHHH 4 22.3333 which is only 3.2% higher than
21.6425, the best known value [2]. REFERENCES [l] 6.O.Hirschfelder. Epstein,
Advan.
W.Byers Quantum
Broxn and S.T.
Chem.
I (1964)
228.
A. Dalgamo, Advan.Chem.Phys. 12 (1967) W3. [3] R. J. Be& Proc. Phys.Soc. (London) 86 (1966) 17_ [4] A.Dalgarno and kE. Kingston, PrOC.ROy. SOC. [Z]
A259 (1960) 424. [5] R. J. Bell and A. E. Kingston, Proc. Phys. SCC. (London) 88 (1966) 901. [S] C. L. Pekeris, Phys. Rev. 213 (1959) i216. [I’] K. R. Piech and J. S. Lewinger. Phys. Rev. 135A
(19G) 332. [8] M. N. Adamov, M. D. Balm&x and T.K. Rebate, Intern. J. QuantumChem. 3 (1969) 13. 191L. Salem. bfoi. Phys. 3 (1960) 441. [lo] C. E. Moore, Atomic energy levels. Circular 467, vol. 1 (1949). Vol. 2 (1958). Vol. 3 (1958) (Natl. Bur. Std., Washington).
259