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Universal force constant relationships and a definition of atomic radius
V&me 10, number 3
CHEMICAL PHYSICS LETTERS
1 August 1971
UNIVERSAL FORCE CONSTANT RELATIONSHIPS AND A DEFINITION OF ATOMIC RADIUS * AB. ANDERSON ** and R.G. PARR Depmmenr
of L‘hemistry,The Johns Hopkins University, Baltimore,Mplyland212.l8, USA Received 2 June 1971
Let k, and R, be the equilibrium harmonic force comtant and intemuckar distance of a diatomk molecule a& where a and p are elements from col-dmns A and B of the periodic table, with atomic numbersZ, and Zs- It is shown empirically that to good accuracy is linear in R,
Mk&$!&
0
and ln(Z&+
is linear in R,
,
im
as a ranges through A or p ranges through B, with the slopes and intercepts of many of the straight lines slrrpw independent of A 9nd B. property I is shown to foUou[ if the electron density of each mo!ezuIe, for R near R,, co= sists of exponentially-decaying and characteristic park which perfectly follow the two nuclei, plus a nonperfectly followkng part which is negligible at each of the nuclear positions and which changes very little 2sR dmges. Property II is shown to folIow if R, is the sum of atomic radii R, and R which are distances fkom nuclei out to the position of minimum electron density along the internuclear axis, and iathec~ofdendtyalongtheavisbasimilar sum of exponentially-decaying, characteristic, perfectly fOuGWing components Empirical relations of the form
ke = M’Z&
exp(--yRs& = GtZ,Z$-
are thereby derived from a physical modeL For the whole periodic table, in atomic units, Cc 0.12, t = 2.0. G =S1.6, q = 0.45. A xet of atomic Iadii is developed.
During the vibrations of a diatomic molecule c@, let the electron density p be resolved into a part pa which
tiere p(a)
Z, is the atomic number of a,and the densities and P,(Q) are valuesat a. The quantityR is the
internuclear distance. As the first step in formulating a phySical model,
perfectly follows nucleus a, a part pp w&h perfectly follows nucleus p, and a nonperfectly following part ~NPF. Then the laplachn of the Born-Oppenheimer vibrational potential energy W, with respect to motion of nucleus a, is given exactly by the equation [I]
I+ET assume thut the integralin tq. (I) may be negIzcted and rhat pNpF(a) = 0. Then we have the Poisson equations.for nuclear vibrations,
p2wMw:
Gz W= 4nZaps(a) and simila.rIy Vi W = 47~2~9~(p) -
‘Q
=
2dW
&
RdR.
(2)
4~~&(a)-pQfa~] -zQ
)
(I)
.’
f A&d by research grants to The Johns Hop% University from the ~ationd Science Fougdation tid the Natlo& : Institutes ofHealth. *b F&s&t add& Chemistry Dep&ei& Indiana Un.+& Qr,Bbdm&$on, Indiana 47401, USA; ’ .:
._
:
Elsewhere (I] we have presented evidence that these equations give good estimates of both harmonic and anbaniionic vikational fo:ce constants-for actual molecuies I-,‘.
Vohme
IO, number 3
LO
CHEMICAL PHYSICS LFLTERS
20
30
Fig_2. See caption to fig. 1. These lines are for the alkali metat halides (la-7b groups) only.
4.0
Reb
Fig. 1. Harmonic force constants, k, for ground state diatomic molecules UJ.~ a3 functions of equiliirium internuck~~ di.+ tances, R,. Also k&Z&$ veaus R, whele Zcrand Zs are atomic rmmbcrsof atoms n and 6. Complete data from tables of Haschbach and Laurie [ 61 and Varshni [7]. Some of the Varshni values are +timates, not experimental values EIeven open triangles, 6, represent akali metal diatom&; 186 dots, q repnsnt hybrids, covalent, and ionic diatomics. Line-sob tained by methodof least squares have,from top to bott0111, dopes of -0.63, -0.38, -1.95, and -2.01, respectively. and intercepts of 1.72,O.lO5,21.1, and 1.55, ail in atomic units
We now assume that pg(or) isa firnctin of the identity of atom fl, that is of Z,, and of R, mad that p&i) isa fin&ion of Z, and R. Then ke = 4nZ~zg[f(R)le
=4nZ~Z&$)
3
(4)
where j^(R) is some function of R We also assume that pa(~) is exponentially decayingatR. Then it follows from eq. (3) that p&3) is too, with the same decay constant:
etc.-Thus we cjbtain * ke = 47QJa[pg(cr)l e = 4+zfi[p,u3)1e
9
and our model gives (3)
where Ipa@)], is the value of the density due to atom 8, at a distance R, from nucleus 8, and b (8) ] e is the tie of the density due to atom cc, at a diPstanceR, from nucleus Q. l
When ZD * Zw the .sec-ondof
eq. (3) may yield poor rn* nitudesGhikshowingcoIrecttren~whichsuggestsiIk chiding add&i& constant multipks in eq. (3). The subsqueni argument in the presfmt pape.rwquld be unaffa%ed. ::
294
k, = 47tC&Z, exp(-lR3
.
(6)
We do not intend to imply that eqs. (5) necessarily rep-
resent actual atomic densities; rather, they represent effectjve a~ornic densities in molecules. Fig. 1 shows that eq. (6) is remarkably accurate for the whok periodic table, with Cw 0.123 and j-w 2.00 in atomic units. (including a&Ii metaj $iatomicsBlters all constants reported in the text slightly.) The latter value cor~~ponds to an dffective hydrogen-like one-
CHEMICAL PHYSiCS LETTERS
Volume 10, number 3
electron density. More accurate fitting of data results if the constants C and 5 are allowed to be functions of the columns A and B to which atoms CYand p belong. This is shown in fig. 2. Still more accurate fitting results if one takes just one cr and lets p span B. This is shown in fig. 3 i-. To complete the model, we assume that R, is the sum of two atomic radii R, and RP which are distances extending out to the point on the internuclear awis at which the electron density is a minimum [3], and we assume that the varibtion of the electron density along the bond wis is given by the sum of characteristic atomic components like eq. (5) each of which is oeponentiah’y decaying with R, with the same decay constant. Then Z, exp(-SIR,)
= Zs exp(-l’RRp) = D
(7)
and Z, Zs = E exp(+r’R,)
1 August 1971
LiF,lJCI.L.iBr.LiI
KF.KCI.KEr.KT
RbF.RbCI.RbBr.RbI CsF.CsCI.CaBr,CsI I 2.0
3D R,
4.0
6,
Fig. 3. Harmonic force constants, k, of alkali metal halide divided by atomic number of the mef.aI. ZM, as functions of equilibrium intemuclear distan~~~~,R,
.
Hence also k, =Fexp[-({-c’)R,]
= Fexp(-r”R,)
(9)
and k, = C(Z, Z,@“~~~
= c(z,Z,)-q
.
m
For the whole periodic table, D = 1.055,6’ = 1.38, E= l.ll,F= 1.72,c”= 0.63, C = 1.64, and Q = 0.45. Fig. 1 shows the test of eq. (9) for the whole table. Eqs. (7)~(IO) can be resolved by Ietting the constants have column dependence, and a final greater resolution is obtained by keeping Q fixed and letting p span its colu-run *. See figs. 2 and 3 and footnote on p. 294. From eq. (7) follows that asp ranges through B with ar fmed, R will be linear in In(ZdZ,), with the value 2U, at the point ZdZa = 1_From such straight lines, we can determine an atomic radius forratom 01for use whenever it is bound to an atom from column B of the periodic table. Fig. 4 tistrates &heprocedure and atomic radii determined in this way are presented in table 1. The radii add to give R, to about the same accuracy t These correlations are documented in detail in ref. [ 21. * If one prefers to adhere the same constants for mokcdes @. q, ~6, etc., as 8, 7,s range through a row in the periodStable, eq. (9) yields the exponential dependence dkcovered by Herschbach and Laurie [4].
Fig. 4. Ratio of atomic numbers for aIkali metal halides as functions of equiiiirium intemuckar distances, R,. The intercept of a line with the ZdZs = t line corresponds to twice the assigned atomic radius of atom 6 or atom (I as appropriate; see text Lines determined by method of least squares.
as provided by the method of Schomaker and Stevenson [S] . For 92 malecuk with known R, that are composed from the atoms in-table I, the average error in predicied R, from our atomic radii is 0.08 A ** +* Dr. R.G.A.R. Mac&an has pointed out to us that more a~curate reIative vaJue.s of R, and RF for purposes ofpredieting internudear distances, may be obtained by simple
least square analysisoP&irs of columns, makinguse of
n0
physical model. 295
~ohun~~lO;immbccr :,
3
CHEMICAL PH~SICS’LETTERS
Table I .’
V&es
‘, Atom
Groups
Li
-., “la-?b
Ni3 K Rb
cs
‘.
;
-4b-7b ..,
@A)
0.55 1.06 1.34’ 151 f.68
F cl
0.94 1.27
Br
1.47
I
1.65
F
0.83 1.40
BI I
In ‘1 Tl
F Q Br
C Si G& Sn Pb
0.54 0.90 1.11 1.28 1.46
0
C Li
0.49 0.91 1.14 1,33 150
& Ga
e0 Sn Pb
1.
S
Se
-Te F cl BI
1 Au&t
1971 ‘.
co& both ioiiic tid‘covalent
species may be tindersfo-dd from-the fact that the oper-
Atom
0.52 0.95 1.18 1.36 1.49
ib-7b ..
That k&correlations R,
R, (A)
0.51 .I Be Ms .'0.99 ca 1.23 ~ Sr1.47 Ba -. 1.61
?a-7b
4b-6b
of atc&mdii
.- ; .’
‘.
‘..
1.61
0.74
_ __
1.05
atcx Vz annihilates coulokbic tegs behaving ai l/R ‘iI+ they cover both sihgle and multiple bonds may be uriderstood from the fact that pi-orb&l contributidns do not’affect p(a). Eq. (3) appears to represent some common essence of all bonds.
Re&sences: [ 1] A.B. Anderson 3375.
[21 RB. Anderson, Thkis, The Johns Hopkins University (1970) [ 31 RFSJ.
1.22 1.37 [4] 0.63 0.94
[S]
1.13 1.29
(61
0.7 1
(71
1.00 1.18
and R.G. Parr, J. Chern. Phys. 53 (1970)
Bader and P.M. Beddzll, Chem. Phys Letters 8 (1971) 2q D.R Herschbxh and V.W. Laurie, I. Chem. fiys 35 (19611458. V. Schomaker and D.P. Stevenson, J. Am. Chem, Sot. 63 (1941) 37. D.R. Htxschbach and V.W_ Laurie, Univ. of California Radiation Laboratory, UCRL 9694 (Berkeley, Calif., 196 1). Y.P. Varshni, J. Chem. Phys 28 (1958) 1081.
CHEMICAL PHYSICS LETTERS
1 August 1971
UNIVERSAL FORCE CONSTANT RELATIONSHIPS AND A DEFINITION OF ATOMIC RADIUS * AB. ANDERSON ** and R.G. PARR Depmmenr
of L‘hemistry,The Johns Hopkins University, Baltimore,Mplyland212.l8, USA Received 2 June 1971
Let k, and R, be the equilibrium harmonic force comtant and intemuckar distance of a diatomk molecule a& where a and p are elements from col-dmns A and B of the periodic table, with atomic numbersZ, and Zs- It is shown empirically that to good accuracy is linear in R,
Mk&$!&
0
and ln(Z&+
is linear in R,
,
im
as a ranges through A or p ranges through B, with the slopes and intercepts of many of the straight lines slrrpw independent of A 9nd B. property I is shown to foUou[ if the electron density of each mo!ezuIe, for R near R,, co= sists of exponentially-decaying and characteristic park which perfectly follow the two nuclei, plus a nonperfectly followkng part which is negligible at each of the nuclear positions and which changes very little 2sR dmges. Property II is shown to folIow if R, is the sum of atomic radii R, and R which are distances fkom nuclei out to the position of minimum electron density along the internuclear axis, and iathec~ofdendtyalongtheavisbasimilar sum of exponentially-decaying, characteristic, perfectly fOuGWing components Empirical relations of the form
ke = M’Z&
exp(--yRs& = GtZ,Z$-
are thereby derived from a physical modeL For the whole periodic table, in atomic units, Cc 0.12, t = 2.0. G =S1.6, q = 0.45. A xet of atomic Iadii is developed.
During the vibrations of a diatomic molecule c@, let the electron density p be resolved into a part pa which
tiere p(a)
Z, is the atomic number of a,and the densities and P,(Q) are valuesat a. The quantityR is the
internuclear distance. As the first step in formulating a phySical model,
perfectly follows nucleus a, a part pp w&h perfectly follows nucleus p, and a nonperfectly following part ~NPF. Then the laplachn of the Born-Oppenheimer vibrational potential energy W, with respect to motion of nucleus a, is given exactly by the equation [I]
I+ET assume thut the integralin tq. (I) may be negIzcted and rhat pNpF(a) = 0. Then we have the Poisson equations.for nuclear vibrations,
p2wMw:
Gz W= 4nZaps(a) and simila.rIy Vi W = 47~2~9~(p) -
‘Q
=
2dW
&
RdR.
(2)
4~~&(a)-pQfa~] -zQ
)
(I)
.’
f A&d by research grants to The Johns Hop% University from the ~ationd Science Fougdation tid the Natlo& : Institutes ofHealth. *b F&s&t add& Chemistry Dep&ei& Indiana Un.+& Qr,Bbdm&$on, Indiana 47401, USA; ’ .:
._
:
Elsewhere (I] we have presented evidence that these equations give good estimates of both harmonic and anbaniionic vikational fo:ce constants-for actual molecuies I-,‘.
Vohme
IO, number 3
LO
CHEMICAL PHYSICS LFLTERS
20
30
Fig_2. See caption to fig. 1. These lines are for the alkali metat halides (la-7b groups) only.
4.0
Reb
Fig. 1. Harmonic force constants, k, for ground state diatomic molecules UJ.~ a3 functions of equiliirium internuck~~ di.+ tances, R,. Also k&Z&$ veaus R, whele Zcrand Zs are atomic rmmbcrsof atoms n and 6. Complete data from tables of Haschbach and Laurie [ 61 and Varshni [7]. Some of the Varshni values are +timates, not experimental values EIeven open triangles, 6, represent akali metal diatom&; 186 dots, q repnsnt hybrids, covalent, and ionic diatomics. Line-sob tained by methodof least squares have,from top to bott0111, dopes of -0.63, -0.38, -1.95, and -2.01, respectively. and intercepts of 1.72,O.lO5,21.1, and 1.55, ail in atomic units
We now assume that pg(or) isa firnctin of the identity of atom fl, that is of Z,, and of R, mad that p&i) isa fin&ion of Z, and R. Then ke = 4nZ~zg[f(R)le
=4nZ~Z&$)
3
(4)
where j^(R) is some function of R We also assume that pa(~) is exponentially decayingatR. Then it follows from eq. (3) that p&3) is too, with the same decay constant:
etc.-Thus we cjbtain * ke = 47QJa[pg(cr)l e = 4+zfi[p,u3)1e
9
and our model gives (3)
where Ipa@)], is the value of the density due to atom 8, at a distance R, from nucleus 8, and b (8) ] e is the tie of the density due to atom cc, at a diPstanceR, from nucleus Q. l
When ZD * Zw the .sec-ondof
eq. (3) may yield poor rn* nitudesGhikshowingcoIrecttren~whichsuggestsiIk chiding add&i& constant multipks in eq. (3). The subsqueni argument in the presfmt pape.rwquld be unaffa%ed. ::
294
k, = 47tC&Z, exp(-lR3
.
(6)
We do not intend to imply that eqs. (5) necessarily rep-
resent actual atomic densities; rather, they represent effectjve a~ornic densities in molecules. Fig. 1 shows that eq. (6) is remarkably accurate for the whok periodic table, with Cw 0.123 and j-w 2.00 in atomic units. (including a&Ii metaj $iatomicsBlters all constants reported in the text slightly.) The latter value cor~~ponds to an dffective hydrogen-like one-
CHEMICAL PHYSiCS LETTERS
Volume 10, number 3
electron density. More accurate fitting of data results if the constants C and 5 are allowed to be functions of the columns A and B to which atoms CYand p belong. This is shown in fig. 2. Still more accurate fitting results if one takes just one cr and lets p span B. This is shown in fig. 3 i-. To complete the model, we assume that R, is the sum of two atomic radii R, and RP which are distances extending out to the point on the internuclear awis at which the electron density is a minimum [3], and we assume that the varibtion of the electron density along the bond wis is given by the sum of characteristic atomic components like eq. (5) each of which is oeponentiah’y decaying with R, with the same decay constant. Then Z, exp(-SIR,)
= Zs exp(-l’RRp) = D
(7)
and Z, Zs = E exp(+r’R,)
1 August 1971
LiF,lJCI.L.iBr.LiI
KF.KCI.KEr.KT
RbF.RbCI.RbBr.RbI CsF.CsCI.CaBr,CsI I 2.0
3D R,
4.0
6,
Fig. 3. Harmonic force constants, k, of alkali metal halide divided by atomic number of the mef.aI. ZM, as functions of equilibrium intemuclear distan~~~~,R,
.
Hence also k, =Fexp[-({-c’)R,]
= Fexp(-r”R,)
(9)
and k, = C(Z, Z,@“~~~
= c(z,Z,)-q
.
m
For the whole periodic table, D = 1.055,6’ = 1.38, E= l.ll,F= 1.72,c”= 0.63, C = 1.64, and Q = 0.45. Fig. 1 shows the test of eq. (9) for the whole table. Eqs. (7)~(IO) can be resolved by Ietting the constants have column dependence, and a final greater resolution is obtained by keeping Q fixed and letting p span its colu-run *. See figs. 2 and 3 and footnote on p. 294. From eq. (7) follows that asp ranges through B with ar fmed, R will be linear in In(ZdZ,), with the value 2U, at the point ZdZa = 1_From such straight lines, we can determine an atomic radius forratom 01for use whenever it is bound to an atom from column B of the periodic table. Fig. 4 tistrates &heprocedure and atomic radii determined in this way are presented in table 1. The radii add to give R, to about the same accuracy t These correlations are documented in detail in ref. [ 21. * If one prefers to adhere the same constants for mokcdes @. q, ~6, etc., as 8, 7,s range through a row in the periodStable, eq. (9) yields the exponential dependence dkcovered by Herschbach and Laurie [4].
Fig. 4. Ratio of atomic numbers for aIkali metal halides as functions of equiiiirium intemuckar distances, R,. The intercept of a line with the ZdZs = t line corresponds to twice the assigned atomic radius of atom 6 or atom (I as appropriate; see text Lines determined by method of least squares.
as provided by the method of Schomaker and Stevenson [S] . For 92 malecuk with known R, that are composed from the atoms in-table I, the average error in predicied R, from our atomic radii is 0.08 A ** +* Dr. R.G.A.R. Mac&an has pointed out to us that more a~curate reIative vaJue.s of R, and RF for purposes ofpredieting internudear distances, may be obtained by simple
least square analysisoP&irs of columns, makinguse of
n0
physical model. 295
~ohun~~lO;immbccr :,
3
CHEMICAL PH~SICS’LETTERS
Table I .’
V&es
‘, Atom
Groups
Li
-., “la-?b
Ni3 K Rb
cs
‘.
;
-4b-7b ..,
@A)
0.55 1.06 1.34’ 151 f.68
F cl
0.94 1.27
Br
1.47
I
1.65
F
0.83 1.40
BI I
In ‘1 Tl
F Q Br
C Si G& Sn Pb
0.54 0.90 1.11 1.28 1.46
0
C Li
0.49 0.91 1.14 1,33 150
& Ga
e0 Sn Pb
1.
S
Se
-Te F cl BI
1 Au&t
1971 ‘.
co& both ioiiic tid‘covalent
species may be tindersfo-dd from-the fact that the oper-
Atom
0.52 0.95 1.18 1.36 1.49
ib-7b ..
That k&correlations R,
R, (A)
0.51 .I Be Ms .'0.99 ca 1.23 ~ Sr1.47 Ba -. 1.61
?a-7b
4b-6b
of atc&mdii
.- ; .’
‘.
‘..
1.61
0.74
_ __
1.05
atcx Vz annihilates coulokbic tegs behaving ai l/R ‘iI+ they cover both sihgle and multiple bonds may be uriderstood from the fact that pi-orb&l contributidns do not’affect p(a). Eq. (3) appears to represent some common essence of all bonds.
Re&sences: [ 1] A.B. Anderson 3375.
[21 RB. Anderson, Thkis, The Johns Hopkins University (1970) [ 31 RFSJ.
1.22 1.37 [4] 0.63 0.94
[S]
1.13 1.29
(61
0.7 1
(71
1.00 1.18
and R.G. Parr, J. Chern. Phys. 53 (1970)
Bader and P.M. Beddzll, Chem. Phys Letters 8 (1971) 2q D.R Herschbxh and V.W. Laurie, I. Chem. fiys 35 (19611458. V. Schomaker and D.P. Stevenson, J. Am. Chem, Sot. 63 (1941) 37. D.R. Htxschbach and V.W_ Laurie, Univ. of California Radiation Laboratory, UCRL 9694 (Berkeley, Calif., 196 1). Y.P. Varshni, J. Chem. Phys 28 (1958) 1081.