E. R. LIPPIKCOTT AND J. 31. STUTMAK
2926
Polarizabilities from s-Function
by E. R. Lippincott and J. M. Stutman Department of Chemistry, University o f M a r y l a n d , College P a r k , Maryland
(Received A p r i l 17, 1964)
Molecular polarizabilities have been calculated from a semi-empirical &function model by using a variational method and &function electronic wave functions. The model enables one to obtain simple expressions for the parallel and perpendicular components and the mean polarizabilities for diatomic systems. Through use of a &function potential network for polyatomic systenis the model has been extended to give polarizability components for polyatomic molecules which can be summed to give the mean values. Polarizability contributions from bond region electrons are distinguished from those due to nonbond region electrons. For polar molecules improved results are obtained by introducing a polarity correction. A comparison of observed and computed mean polarizabilities indicates that the results are quite reasonable with the average deviation being 10% for the 55 molecules studied. The approach used has the advantage that calculations can be made on large molecular systems without undue computational difficulties. Bond polarizabilities are not necessarily transferable from one molecular system to another owing to the strong dependence of the bond parallel component on bond length and the effect of configuration on the summation rules.
Introduction The electrical polarizability is one of the fundamental physical properties of a molecular system being the microscopic analog of the index of refraction, a macroscopic bulk measurable. These are linked by the well-known Lorentz-Lorenz equation
(--)(+) +
n2 - 1 4 / 3 ~ =N ~n2 2
(1)
where CY is the mean molecular polarizability; n, the index of refraction; N , Avogadro’s number; d , the density; and M , the molecular weight.* Many calculations in recent years have been made in the attempt to obtain atomic and molecular polarizabilities from quantum mechanical models. I n 1930 H a ~ s obtained e ~ ~ polarizability values for the helium atom and lithium ion. He used a variational method in testing several types of ground-state wave functions together with several types of perturbed state wave functions in terms of the perturbing potential. Bucki i i g h a m ’ ~approach ~~ is based on Kirkwood’s3c variation method and heavier atoms are considered, leading to complex determinantal self-consistent field wave functions. Bell and Long4 compare polarizability T h e Journal of Physical Chemistry
values obtained from six different unperturbed wave functions chosen for Hz+ and H z molecules and conclude that the polarizability is not very sensitive to the wave function chosen for the unperturbed molecule. The calculated values agree with experiment, 15% deviation being the inaxinium for any wave function chosen and about 3y0being the minimum Abbott and Bolton6 studied X2and used the polariz(1) (a) Taken in part from the thesis submitted by J. M. 8. in partial fulfillment of the requirements for the degree of Doctor of Philosophy, University of hfaryland; (b) this work was supported in part by a grant to the University of Maryland for a Materials Science Program by the Advanced Research Projects Agency, Department of Defense. (2) The polarizability is strictly a tensor
(.. . a,,
a =
0l.Z
ff*y ayy
ffzy
z) 0121
and the value of u i j gives the magnitude of the dipole induced in the i direction by a unit electrical Aeld in the j direction. However, the mean molecular polarizability obtained experimentally is the average of the three diagonal components. (3) (a) H. It. Hasse, Proc. Cambridge Phil. Soc., 26, 542 (1930); (b) R. A. Buckingham, Proc. R o y . S O C . (London), A160, 94 (1937); (c) J. G. Kirkwood, P h y s i k . Z., 33, 57 (1932). (4) R. P. Bell and D. A. Long, Proc. R o y . S O C .(London), A203, 304 (1950). (5) J. A. Abbott and H. C. Bolton, J . Chem. Phys., 20, 762 (1952); Proc. R o y . SOC.(London), A216, 477 (1953).
POLARIZABILITIES FROM &FUNCTION POTENTIALS
ability as a criterion for determining the molecular wave function of a system by a self-consistent field method and imply that the polarizability is more sensitive to the wave function chosen than other measurable physical properties. This is just what one would expect from a conceptual frame of reference, Le., t h a t the polarizability is a measure of the ease with which the electron probability distribution may be distorted and thus would be a sensitive indicator of the correctness of the wave function used. Kolker and Karplus6 have recently presented the first attempt a t the calculation of the polarizability tensor a with ab initio wave functions for a series of first row diatomic molecules. Their results again indicated that the polarizability could be a useful criterion for te9ting the accuracy of wave functions. Essentially no work has been done on more complicated systems. The d-function model of chemical binding is useful in predicting vibrational energies of both diatomic and polyatomic systems and also in generating a n internuclear potential function. Lippincott and Dayhoff using this semi-empirical method predict W e , ueXe,De, and re for both diatomic systems and bonds of polyatomic molecules. In view of the studies of Abbott and Bolton and more recently of Kolker and Karplus that the polarizability is an effective criterion of how well the wave function approximates the real situation and since the 6-function model performs reasonably well in the prediction of the other molecular constants, we present this investigation as an attempt to test the acceptability of the d-function wave functions with respect to calculations of electric polarizabilities.
The &Function Model It will be necessary to review briefly those aspects of the &function model of chemical binding which have application to the calculations of polarizabilities for molecular sy~tems.’,~ The potential energy for the n-electron problem is taken to be the sum of single &function potentials each having the following form for a diatomic system
where x is the coordinate of motion along the internuclear axis, a is the b-function spacing, A I and A z are the &function strengths or reduced electronegativities (REX) for nucleus 1 and 2, respectively, g is the unit d-function strength (the values for the hydrogen atom), and 6 ( x ) is a &function whose properties are
6(x)
=
0 when x # 0
(3)
2927
s-:’
6(x)
whenx
=
6(x) dx
=
=
0
(4)
1
(5)
for any argument x of 6. Thus for the case in question the potential is zero everywhere except a t the &-function positions, Le., x = a / 2 and x = -a/2. The REN or A values are, in principle, obtainable from separated atom energies and the equation
A
=
4-2E,
(6)
which is obtained from the solution of the atomic problem using the b-function model.g For molecules involving other than hydrogen atoms, other approaches for determining the R E X values were needed. Empirical rules were developed and in Table I the R E N values of the elements are listed. The solution to the Schroedinger equation for the molecular problem has the form
and E , is the separated atom energy for the ith particle. In the above equations, the positive sign corresponds to a bound state and the negative to a repulsive state. See Fig. 1 for an illustration of the probability density molecule. function for both states of the Hz+ Frost’s* d-function branching condition can be used to obtain an expression for c as a function of the internuclear distance. a
2+‘
fi‘
(6) H. J. Kolker and M. Karplus, J . Chem. Phys., 39, 2011 (1963). (7) E. R. Lippincott and LM. 0. Dayhoff, Spectrochim. Acta, 16, 807 (1960). (8) A. A. Frost, J . Chem. P h y s . , 2 2 , 1613 (1954); ibid, 2 5 , 1150 (1956): ibid.,23, 985 (1955). (9) One of the better empirical equations for the reduced electronegativity is
A = [X/(2.6m - 1 . 7 ~- 0 . 8 0
+ 3.OF)j’/z
where X is the electronegativity of the atom taken from Pauling, n is the principal quantum number: p is 1 for atoms with p electrons in valence shell, 0 for atoms with no p electrons in valence shell; D is the total number of completed p and d shells in atom; F is the total number of completed f shells in atom.
Volume 68, Number 10
October, 1964
E. R. LIPP~NCOTT A X D J. ILL STUTWAN
2928
and
The model can be extended t o polyatomic molecules by forming linear combinations of atomic d-function wave functions and by making use of the branching condition.8 For example, if we consider a triatomic molecule as in Fig. 2, and allow the molecular wave
-l 9
-5-€
=
-2.49
These two conditions give solutions for the homonuclear case of the form c, = A g [ l
+ e-aci]
(12)
By combining eq. 12 with 6 we have lim c, a-
=
Ag
=
V'-2E,ia=m,
m
Lippincott and Dayhoff' by-pass the problem of obtaining the c , for each individual electron by generating a "super)' one-electron situation from the
Figure 2 . &Function network potential system for a bent triatomic molecule.
function to have 2n branches a t each atom involved in n bonds, the molecular wave function is
9
= a191
+ a292 +
a3$3
(16)
providing that the atomic wave function is defined along the given branch. Thus *I
= ' k I V = a191
= *I1
+
and *I11
= *v
= *VI
= az+2
Now using
+,= e-c"xb' where b refers to the bond coordinate, we have for the regions indicated in Fig. 2
Figure 1. &Function electron probability density as a function of position for H2+ (unnormalized).
corresponding n-electron situation. A resultant c is obtained (written CR) and is assumed to account for an electron pair. For the homonuclear situation CR =
Ad;#
=
V ' C Z
T h e Journal o j Physical Chemistry
=
aleCix
ale-C'"
+
azec2(z
+
=
azec2U
=
,ze-c2U
- Rd
azec2(x
- Rd
- HI)
aze-C2(Z
+ a3ec3(~-
$v = u 2 e - w+
+
a3ec"'"
a3e-c3(Y
'2)
- Rd
- Rz)
(20) (21) (22) (23) (24) (28
At branching points 1, 2, and 3, there are two, four, and two branches respectively. The &function condition gives
( d n l n z ~ l ~ z ~ l (15) ~z)'/~
and then solving the wave equation as if the molecule were homonuclear.
$11
@JIr
___CR,2
=
$Iv = ale-"'z +
(14)
where A is the one-electron d-function strength for the atom, n is the principal quantum number, and N is two tiines the column number in the periodic table. Heteronuclear diatoniics are readily treated by forming a geoiiietric mean molecular &function strength C R ~ ~
$I
V l I ( Z . ~
+
t- +'IrI>.~+-+:v~:) 4hr
$'Iv(zr)
where p i j = e'''''. Secular eq. 30 and 31 when exparided into eleiucnt-cofactor forni give the following two-by-two determinants
x = Ri o r y = o
and
where (sf) iniplics the derivative is taken from the nucleus to 5 > 0, ie., d ) s : ( / d s= 1 ; similarly (s$) implies dls l/ds = - 1. Because of the presence of four branches a t atom 2 , ey. 26, 27, 28, and 29 give two three-by-thrce detcrminantal solutions
I
0
0
(1 -
respectively. These solutions are identical with thosc which would bc obtained if the triatomic iiiolecule were coiisidered to consist of two diatomic nioleculcs. This ciiwrnstance is a by-product of allowing the atomic &function wave fuiictiori to cxist only along the 'internuclear axis involved and allowing the axis to extend infiiiitcly in both directions, by prohibiting interaction of neighboring bonds. When eq. 30 and 31 are considered in detail we find that the two reprcsciitations are generated because of the two fornis which the wave function can take a t nucleus 2 . Of course, thc values of these two foims must bc identical a t this nucleus. Of greater eignificance, however, is the fact that the secular deterniinant can takc on a different forin for each form that the common $ can assunic and thus the method is general for any niolccular system with localized bonds. For example, one of these determinants for methane is given in eq. 34.
: ) ? I * *
(30) and 0
1 I
I
1
0
0
0
0
I
0
0
0
I
0
L
I
1
1
(1 - ;)PIl
I
I
I =
i i
'
1
0
I
0
1
0
0
(1 - 2 Q ) P I 4 '
o
(34)
E. R. LIPPINCOTT AN) J. -\I. STUTMAN
2930
Kote that the determinant given in eq. 34 reduces to
where p , , = eczR7where R3 is the internuclear distance 1). between atom 1 (the carbon atoni) and atom ( j The other three determinants are the saiiie as I esccpt for the changing p , , and the position of the 1 in the first row.
+
Lippincott and Dayhoff' further consider heteronuclear bonds in polyatoniic molecules as pseudohomonuclear and treat theiii by the saiiie method as were similar bonds in diatoniics. Thus thc determinantal eq. 32 and 33 are changed so that
to both the parallel and perpendicular components, and the method by which these bond components are summed to give the average inolecular polarizability. General Approach. Hirschfclder, Curtiss, and Birdlo" discuss a variational trcatinent first introduced by HyllcraaslOb and We use this method here with the perturbing potential H , taken as
HI
=
-d&&
[- 5 + 2 Z a X a ] 5,
r = l
(38)
rr=l
If there are n equivalence classes of electrons, the equivalence being based on identical radial wave functions, z t is the coordinate of any one of the (Aleetrons which falls in the zth equivalence class. I n eq. 38 A is the reduced electronegativity (REKj (in atoniic units) of the nucleus (if it is an atom which is perturbed) or the root mean square of the reduced electroncgativities of the two nuclei involved in a bond (if it is a bond which is perturbed), g is the unit d-function strength, E, is the electric field coniponcnt in the 5 direction, i is the clectron equivalence class index, and CY is the nuclear index. One must use Ag in this equation rather than e because of the fact that in the niodel used the equivalent of Ag is e'. Thus an equation for cyzz is generated
4nA
azz = __ a0
__ [(XI
- 2)Z -
(n -
l)(Sl - Zl)(XZ
-
$12
(39)
where A,' is the geometrical mean of A i and A,. This approxiiliation does not contribute significantly to the polarizability calculations since the tcrni involving the &function strengths is very sniall.
The &Function Model Applied to Molecular Polarizabilities
where z1 is the coordinate of any electron in the first equivalence class and 2 is the average coordinate of any one of these elcctrons. It is to be noted that (21 - ~ I ) ( X Z- 5,) = 0 since the &function wave functions allow no interaction between coordinates. The model with the mean d-function strengths predicts 2 = 0 so that eq. 39 becomes
Since CY,
=
l/dCYl
+ ffz + 4
(37)
where al,CY%, and cy3 refer to the three principal polarizability components. one niust havc a method of grncrating component polarizabilities in order to conipute molecular (average j polarizabilities. I t is necessary to build thc iiiolecular values froni a consideration of the bond and atoniic polarizabilities and to dctcrinine sonic1 suinniation rules. I n this section we will discuss the calculation of parallel bond coiiiponcnts froiii thc niolecular d-function niodel, the gciicration of bond perpendicular components froin atoiiiic &function polarizabilities, corrections to the parallel component due to nonbond-region electrons, polarity corrections The Journal of Physical Chemistry
or equivalently
whew n is the nuiiiber of equivalence classes. For instance, if there are four electrons in the bond region, two foiwing a 7r-bond, and two forming a u-bond, there are two equivalence classes of electrons and n = 2. The implied assuiiiption is that there will be as (10) (a) J. 0. Ilirschfelder. C. F. Curtiss, and 11. B. Bird, "Molecular Theory of Gases and Liquids," John Wiley and Sons, Inc., Sew York, PI'. Y., 1954; (b) E. Hyllerans, 2. Physiic, 6 5 , 209 (1930).
POLARIZABILITIES FROM ~ - ~ ~ L W C T I OPOTENTIALS N
much a contribution to the bond coiiiponcnt of the polarizability froin a u-bond as there is froni a T bond. This, of course, is oiily a n approximation to the real situation. In the case where no reasonable single structural form can be written for the niolecule, n is talcen to be the bond order, e . g . , in SOZ, n for the S-0 bond is 3 / z . For a t o m in their ground state, n = 1, i.e., all electrons in the valence shell of an atom are considered to have nearly the sanie charge distribution so t h a t they would all belong to the saiiie equivalence class; the atomic polarizability problcni reduces to a one-classelectron problem. Thus the procedure is to generate values for ?x for the system using the &function wave functions and the einpirical &function strength (reduced electronegativitics) obtairicd by Lippincott and D a y h ~ f f . ~Once this expectation value is obtained, it is used in eq. 40 to obtain the desired polarizability component. Atomic Polai.izabilzties. To obtain x-~ values for atoiiis,-we assunie the atom is isotropic and that 2 = ._ 2 = 22 = r 2 / 3 ; we calculate F2 using the &function located a t the nucleus. Thus in polar coordinates” $ = Ne-’‘
(42)
Normalization gives
N
=
A~I’z~-
(43)
Further, the expectation value of r 2 is r-5
=
lT~z’~m \Er2Wzsin 0 ded$dr = 3 / A z
(44)
thus
Table I : Reduced Electronegativities“’* IA
IIA
IIIA
=
1/A2
(45)
Solution of the Hamiltonian for negative l? gives
‘Thus A = 4% is the sanie &function strength of the atoiii obtained either froin the first ionization potential or from the REX values given in Table I. l h a l l y , the polarizability along the x-axis of an atom is a,, =
4 --
ao3A3
(47)
Table I1 is a conipilation i n periodic chart forin of atoniic polarizabilities calculated froni cy. 47. 1Iany expcriiiiental values have been calculated froni honionuclear niolccules on thc basis that theine are no bond coinponents, e . g . , the average polarizability of the chlorinc niolcculc C Y C Iis~ given i n Batsaiiov’s12atext as
IVA
VA
VIA
VIIA
VI11
H
He
1
1.00
2
0.439 0.538 0.758 0.846 0.927
3
Na hlg AI Si P S C1 Ar 0 . 3 5 8 0.414 0 . 5 3 3 0 . 5 8 3 0 . 6 3 0 0.688 0 . 7 5 3
4
K Ca Ga Ge As Se Rr Kr 0.302 0.337 0.472 0.536 0.564 0.617 0.633
5
Rb Sr In Sn Sb Te I Xe 0.286 0.319 0.422 0 , 4 8 3 0 , 4 9 6 0.538 0.584
6
Cs Ba T1 Pb Hi Po At Rn 0.255 0.289 0.320 0.352 0.365 0 , 3 9 9 0.421
Li
Fr
Be
B
C
N
O 1.00
F
Ne
1.065
Ra
7
’ In a.u. T h e REP4 values tabulated here are the A values of ref. 6 in the text. twice the average Polarizability of the chlorine atom LYCI. As Denbigh’2h points out, perfect additivity could only occur if all the atonis in the niolecule were entirely without effect on each other, so t h a t no bonds were formed. Thus, it is inaccurate to assunie t h a t atomic polarizabilities sum up t o equal the niolcculai polarizability, and it is likewise inaccuraie to assunie the molecular polarizability could be broken down into only atoiiiic coiiiponents. A diatoniic niolecule has an axis of sylnnietry so t h a t QM
__ 2 2
293 1
=
‘/8(al/
+2 a ~ )
(48)
where ai,is the polarizability along the bond and aI is the polarizability perpendicular to the bond. Denbigh’2ha’dcalculates these coniponents knowing two of the three expel-iniental quantitics, the Kcrr constant, the refractive index, arid thc depolarizatioii ratio. One finds that the perpendicular components which he obtains are qualitatively equal to the sum of the rcspective atomic polarizabilities calculated from eq. 47. Whenever possible thei-efore, we have taken the perpendicular coiiiponents of Deiibigh to obtaill experinieiital values with which to conipare our atoinic polarizabilities (see Table 111). The ratioiiale behind (1 1) The wave futictioti corresponding to n one-dimensioiinl model i n Cartesian coordiii:ites. $ = Nr-c(l1.1+2/+lz ) , would .not give the s ~ ~ l i e t ~distribution i~~zil which is expected. (12) (:I) S. S. R:itsoiiov, “l for a series of molecular Table VI11 shows a
Figure 7 . Bonded triatomic system.
Consider, now, the same syst,ein but wit,h bonds between ,4and B and between l3 and C (Fig. 7). It is seen that one degree of atoiiiie “polarizability” freedom is taken up per bond per atoni, i.e., AB i n Fig. 7 rcplaces one CYA and one QB in Fig. 6 ; thus a h 1 = ‘/3(QiIno
+
cy,lllc
+ +2aA f QB
2aC)
(77)
or cyM
+
=
1/3(al!’,,% \
where the first attenipt a t
%c
+c
“1,)
(78)
z
2a1, is written as a
c 2Q.L c (3 =
a
-
J
4%
(79)
where n j is the nuiiiber of bonds in which tho j t h atom is involved. For nicthanc this tcriii would he Z ~ C Y=~8 ,~ r f I-
systems calculated from eq. 81, together with q,, a M (ealed.), and the corrcsporiding experiiiientally detcrniined ~ $ 1 . The agreement for nonpolar niolecules is quite satisfactory. Howcver, it is to be notcd that in general the deviation of the calculated values froin expel iniental values is greater for polar niolccules. Thus a polarity correction is probably neccssary if this iuolecular polarizability treatment is to be useful for a wide range of molccular systems. Polarity Corrections. T h e sun1 Z 2a1, of the perpendicular components in thc prcvious section was calculated in such a way as to weight all atonlie polarizabilities equally. However, where bond cllectron density asymnietry is believed to exist, the perpendicular component should contain a greater contribution from the atom which would tend to have the larger portion of electron density in its vicinity. This is easily done by wcighting thc atomic contributions to the total perpendicular coiiiponcnt according to the squarc of the electronegativity of the clcrncnt. Thus eq. 81 must be replaced by
O(C
Equation 79 thus appcars to weight very heavily exterior atonis of a iiiolcculc, giving a ricgativc contribution to central atoiiis. This is pi,oba,bly not the case. An alternativc approach is t’o weight, the atoiiiic eontributions ove~’the total reiimining atoiiiic dcgrces of frecdoni, e.g., for the ~iiolcculcAUC
where X , is the clcctronegativity of thc j t h atoni. ‘l’hc form of the fiinct,ional dcpendencc on olcctroncgativity which we have chosen is partially just>ificdhy thc fact that the accepted defiiiition of polarity rclates to thc square of the electronegativities. l 7 A polarit,y corrcct’iori is siiiiilar!y inadc to the individual bond parallcl compoiients hcforc ineorpoi,ating t.hr noribondcd valcncc elcctroii coixction and bcforc sunuiiing up to obtain a l ,. ,Just as iii t,hc diatomic situation, us(: is iiiatlc of I’auling’s defiiiit.ion of the pcr cent ionic charactcrI7
and the per cent covalent character
Polarizabilities ( N o Polarity Corrections)“
0 62.80 72.07 68.85 141.30 46.26 90.67 155.71 52.51 98.32 57.98 69,13 80.27
Carbon compounds 34. 19 0 11.39 11 06 0 34.07 33 30 39.41 0 42.14 42 60 54.36 67.94 0 45.60 44 70 109.63 103 20 187.59 0 29.88 2 . 9 7 26.37 25 90 104.05 0 64,90 62 9 218.67 0 124.79 115 68 47.84 3 . 9 5 34.77 32 3 177.97 7 . 8 9 9 4 . 7 3 84 08 61.47 11.90 43.79 45 60 89.77 23.79 60.90 64 80 119.65 35.68 7 8 . 5 3 82 30
Polyatomic hydrides 24.96 28.05 0 19.06 16.41 2.97 14.18 8.99 3.94 50.91 77.74 0 56.97 83.89 0 50.08 27.08 14.62 30.59 1 6 . 8 3 39.67 61.80 40.39 46.84 37.70 29.60 74.99 87.03
hlolecules with polar bonds 49.49 38.01 1 2 . 6 0 33.37 41.13 56.97 1 6 . 8 0 38.24 69.27 8 6 . 2 4 1 6 . 8 0 57.44 8 1 . 3 1 112.91 16.80 70.34 64.22 111.49 21.00 6h.57 6 1 . 2 0 146.57 25.20 77.66 7 0 . 2 5 181.73 25.20 92.39 8 6 . 6 8 238.84 25.20 116.91 2!)4,74 134,33 23.79 150.95 102.08 35.68 7 8 . 1 3 ! ) 6 63 , !)1.41 149.90 47.58 96,30 47,58 125,24 1 l!). ,55 208.58 194.63 243.90 47.58 162.03 47 58 136.92 131.60 228.57 151.31 323.49 4 7 . 5 8 174.13 5 9 , 4 8 223,:38 157.39 453.29 :116,86 152,67 33.27 167,60 12R.69 123.08 49.91 99.56 66.55 157.48 150.53 255.37 182.28 375.01 66.5.5 207 94 358, 10 1!)2,29 50.96 200,44 240. 02 492. 83 101 . 00 278. 25
26.0 21.45 14.44 43.39 49.66 36.40 45.73
h C C
and the parallel coiiiponent of the polarizability must be multiplied by this factor ( u ) in order to compensate for the lower effective expectation value of x 2 , ie., the electron will be found in a region of +space nearer to the atom with the higher electronegativity. Thus the corrected parallel coniponerit is
C
c c C
d C
d c C
C
c d d b b d b
where ut is the per cent covalency and all, is the bond parallel component of the polarizability of the ith bond. Q, is the atoiilic polarizability of the j t h atom and J, is the fraction of the electrons of the j t h neutral atom which are not involved in bonding according to the LinnettI4 picture. The resulting expression for the average molecular polarizability is
r
I
23,82 28.57 33.01 36.46 36.46 44,78 52.47 58,73 90.67 80,05 102,56 111.76 141.00 121,23 137.08 156.78 ll5.!)G 114.13 156. 14 189,07 164.74 277,Sl
h b b b b b b b b b b b b b b b b b b h b h
X 1025~ r n . ~ . S. S. Batsonov, “liefrartornetry and Chemical Structure,” translated hy P. 1’. Sutton, Consultants Bureau, New York, S . Y . , 1!)61. J . 0 . Hirschfelder, C. F. Curtiss, and J j . B. Bird, “Xlirlewlar Theory of Gases and Liquids,” John U’iley and Sons, I n r , , New York, N. Y., 1954. d E . A. Moelwyn-IIrighes, “Physical Chemistry,” Pergamon Press, New York, N. Y . , 1!)61.
where i is the summation index for bonds and j is the summation index for atonis. I n calculating Q M for certain niolecular s y s t e m there is one further point to consider: the term in eq. 86, (3N - 2 n b ) , which is supposed to give the number of remaining atoniic degrees of polarizability freedom. Clearly this number is 7 for methane, 5 for water, etc. ; but when linear or planar nioleculcs are considered, the number of degrees of freedom increases. H C N , for exaniple, has two bonds and three atoms and would be predicted to have five atoniic “degrees of freedom” remaining. Fig. 8 shows, however, that since the niolecule is linear, the carbon atom does not lose a second degree of freedom since the second bond is contained within the same degree of freedom as the first. Siniilarly, BF3 is B
‘I
?’he .lortrnal of Phusical C‘hernistril
Figure 8. Residual atomic polarizability degrees of freedom for H C N .
I’OLAI~IZABILITIES FROM CI-FUSCTION POTESTIALS
2939
Table IX : Molecule
Carbon compounds C (graphi te) C1H2 C2H4
CiHe CeHs HC N C3Hs C6Hl4 CHaOH Dioxane CHrCI CH2Clz CHClr
0
62 20 71 40 68 29 139 94 45 80 89 88 154 34 51 60 95 92 60 31 72 47 83 26
34 I 9 33 41 54 36 6’7 94 187 59 28 47 103 76 218 23 42 45 161 60 55 12 77 53 101 59
‘0
0
0
8
0 0 0 2 97 0
0 3 7 11 23 35
95 89 90 73 68
10 10
18 6 13 36 8 14 7 7 7
11 33 41 45 109 25 64 124 32 88 42 57 73
39 87 92 41 18 75 55 19 67 47 44 93 51
11 06
33 42 44 103 25 62 115 32 84 45 64 82
30 60 70 20 90 90 68 30 08 60 80 30
b c c c
c
c c d c d c c c
where ndl is the number of residual atomic degrees of freedoin. I n Table IX are found U ~ O ~ ! ~ , f, 5 ,
ndl, c ---, O5X5? and c X,? niolecular systems. a5,
Si“ GeHI SHZ SeHl
46 38 29 57 59 65 49 58
51 30 60 46 39 83 01 46
28 05 15 75 7 37 61 79 63 74 71 31 27 0 8 39 99
0 2 97 3 94 0
0 0 14 62 16 83
7 6 5 8 7 7 5 5
24 19 13 39 41 45 30 38
85 01 64 75 04 71 24 43
26 00 2 1 45 14 44 51 12 43 39 49 66 36 40 45 73
c d d b b b d b
OM
3
for a random assortment of
S a m p l e Calculation: CF,
R
=
1.317
x
cm., Ac
=
0.846 a.u., A F = 1.065
a.u., X C = 2.5, X F = 4.0
Polyatomic hydrides CHI NHz OH1 nlH6
3
u
= 0.570
CCF
= 4.37 a.u.
oc = 9.78 X OF
__
zCF2=
R2 -
4
= 4.90 X 10-2scm.3
ao2 += 0.4409 X 2c2
10-l6 cm.2
Molecules with polar bonds 39 37 42 45 47 51 54 57
RFa CFI SiFI GeF4 PFa S Fe SeF6 TeF6 HgCIz
180
CClr SiClr TiClr GeClr SnClr SbCla I-IgBrz BI3ra SiBrr Snllrr HgIz SnI4
96 92 106 125 111 119 135 219 329 142 157 285 225
nc1,
08 41 93 90 24 59 71 51 05 88 91 41 83 38 52 86 41 94 73 60 92 69
14 32 25 33 44 83 96 95 99 79 125 146 139 160 226 335 125 104 I99 292 174 433
06 47 87 87 60 55 31 54 40 62 92 01 02 00 44 44 19 62 19 51 98 69
12 16 16 16 21 25 25 25 23 35 47 47 47 47 47 59 33 49 66 66 50 101
60 80 80 80 00 20 20 20 79 68 58 58 58 58 58 48 27 91 55 55 95 90
7 7 7 7 8 9 9 9 6 7 7 7 7 7 7 8 6 7 7 7 6 7
21 28 28 32 37 53 58 59 101 70 88 100
104 106 131 176 125 94 136 172 170 253
91 89 53 21 61 45 74 42 08 73 80 00 14 32 18 92 96 82 16 22 61 76
23 28 33 36 36 44 52 58 90 80 102 111
141 121 137 156 115 114 156 189 164 277
82 57 01 46 46 78 47 73 67 05 56 72 00 23 08 78 96 13 14 07 74 81
6 b h 6 b b b
* S.S. 13atsonov, “Refractometry and Chemical a x 1026 cm.3. Structure,” translated by P. P. Sutton, Consultants Bureau, J. 0 . Hirschfelder, C. F. Curtiss, and New York, S . Y., 1961. I
