R. A. BONHAM
856
Some New Relations Connecting Molecular Properties and Electron and X=RayDiffraction Intensities1
by R. A. Bonham Department of C h i s t r y , University of Oslo,Blindern, Norway, and Department of Chemistry, Indiana University, Bloomington, Indiana (Received August 1 , 1966)
New integral relations connecting certain molecular properties with scattered electron and X-ray intensities from molecules are presented. The relationships connect molecular expectation values of various powers of the electron-nuclear and electron-electron distances, electronic charge densities at nuclear positions, and the molecular electron density with potentially observable diffraction intensities. A brief review of previously derived relations connecting diffraction intensities with molecular quantities is given.
A number of elegant integral relationships connecting certain observable quantities obtained from cliff raction experiments with molecular properties have been given by Tavard and eo-workers2 and by Silverman and Obata.2 Using the notation introduced by Tavard, Rouault, and Roux2b it is possible t o summarize these relations as follows. The total and elastically scattered intensities for electrons (ED) and X-rays (XR) scattered by molecules may be written in the first Born approximation as
and
where j&) is the zero-order spherical Bessel function sin z/z, I stands for intensity, the subscripts T and e for total and elastic, 2, is the atomic number of the kth atom in a molecule containing N atoms, and r,, is an internuclear distance. The brackets ( )vib stand for vibrational average and the terms IoED and IoXR are constants characteristic of the experimental method employed. I n the electron diffraction case IoED can usually be selected by matching the experimental intensity with the theoretical Coulomb intensity at a large value of the scattering variable s(s = 4a sin (8/2)/X). This is equivalent to selecting IoED (1) Contribution No. 1427 from the Chemical Laboratories of Indiana University; the author wishes t o thank the John Simon Guggenheim Foundation, The Royal Norwegian Council for Scientific and Industrial Research and the National Science Foundation for their financial support. (2) (a) C. Tavard and M. Roux, Compt. Rend., 260, 4933 (1965); (b) C. Tavard, M. Rouault, and M. Roux, J . Cham. Phys., 62. 1410 (1965).
(3) J. N. Silverman and Y. Obata, J . Chem. Phys., 38, 1254 (1963).
The J O U T Wof~ Physical chemistry
MOLECULAR PROPERTIES AND ELECTRON DIFFRACTION INTENSITIES
857
so that U T ~ ~ ( Sand ) ueED(s)go to zero a t some large value of s (ie., s 2 30 for 40-kv electrons and molecules possessing only atoms in the first row of the periodic table). In eq 1-4 the electron densities are defined as
Dn(r) = r2.fdQp(r
+ rn)
P(r) = r2SdQpc(r)
(6)
and
E(r) = r2Sdr’p(r’)SdQp(r
+ r’)
M
S d n . . .,fd7M:IQe126(r - rt)
i=l
i
where i, j refer to the nuclei and 1.1, v to the electrons, or finally2*
(7)
where p(r) =
(8)
and
Before performing the same type of analysis on the other diffraction quantities it is helpful to introduce notation for the various constituent parts of the potential energy as Unn
ztz, -
= i
~ e ” r M/ 2C(Z?
(5)
n-
1
dsIeXR(s)/IoXR
(21)
(4) L. 8.Bartell and R. M. Gavin, Jr., J. Am. Chem. SOC.,86, 3493 (1964). (6)R. A. Bonham, J. L. Peacher, and H. L. Cox, J . Chem. Phya., 40,3083 (1964).
Volume 71, Number 6 March 1967
858
R. A. BONHAM
where strictly speaking all potential energies are vibrationally averaged and the terminology Coulomb and exchange are adopted by analogy with HartreeFock usage, although here the relationships refer to the exact wave function for the system. Silverman and Obata3 have pointed out the existence of eq 21 as well as the fact that eq 15 is also closely related to the nuclear diamagnetic shielding constant which is of interest in nuclear magnetic resonance (nmr) work. The remaining relations are due to Tavard and co-workers.2 Perhaps more importantly, Tavard6 and later Iijimd have shown that by using eq 18 it is possible to derive a formula for the total molecular binding energy, E b , with the aid of the virial theorem. The total potential energy is given by the molecular virial theorem
U(XJ
=
2E(XJ
+ Cxt.VtE(X2) z
=
2.F(Qi)
1 21r
-
[d s [ S ' I ~ ~ ~ ( s ) /-I o ~ ~ - C&wa((na>T + ..
s41AED(s)]
(22)
+ C Q ~ * V ~ E (+QCX;*V,&(X$) Z)
Cfiwt((nt)T f
+
a
allows the vibrational average of eq 23 t o be simplified to (c)vib = 2Eeq 2Cka(Qa9QJvib (24)
+
i
harmonic vibrational energy for the a t h normal mode it may be replaced by the sum of the harmonic vibrational energies in terms of the normal frequencies w,, so that the total electronic energy in the equilibrium state becomes approximately =
'/Z(U)vlb - Cfiwa(na a
+ '/z)
(25)
The last term in eq 25 is simply the well-known zeropoint vibrational energy correction. It is now possible to write the binding energy as The Journal of Physical Chemistry
(27)
-
E( [Zk - F k ( S ) I 2 + sk(s)] k N
s41AED(s)=
+C N
s411AMED(s) =
s41AED(s)
N
[Z,
z#j=1
- Fi(s)] x
[z, - ~ , ( s ) ]e-zi"a''2
(28)
sin srti/src,
I n addition, the second integral in eq 27 reduces to N
d5 [S411AMED(S)
'/2
N
- s41AED(s)] = '/4C C Z J j x tZ.i-1
r,, :
CC f
Xkj2
yzdyk~[Xzi' exp(-- rt,XI"
+ Ets2~tiz/2)-
x",I,
exp( - r i j h 5
+ 1tj2Xk1'/2)I/ [(X1"'
-
b%-til}
(29)
where the term in brackets becomes
a
where the k,, are force constants and Eeq is the equilibrium electronic energy. Since Cka(Qa*Qa)vibis the
Esq
'/2>
because the integrand of the first integral converges to zero a t small values of s (s 15) for molecules containing only light atoms. I A ~ and ~ ( I~I A ) M ~are ~(s) given theoretically as9
(23) where the last term on the right has been left in terms of mass-weighted Cartesian displacement Coordinates and X," is the vector specifying the equilibrium position of the Zth atom. The vibrational average of the term vJi'(X,) is zero at the equilibrium state (Le., (bV(X,)/bXiz)vib, i = 1, 2, 3 are all zero). The energy in normal coordinates through quadratic terms can be written as E(&,) = E,, '/2 Clca&,.&, which
(26)
where ( n , ) ~is the thermal average over vibrational states and s 4 1 A E D ( 8 ) is the theoretical atomic background obtained by letting r f j go to infinity in eq 12. It is more convenient to write eq 26 as6,?
B
a
'/2>
i
where the sum is over all the nuclear Cartesian coordinates.8 This expression can be transformed successively from Cartesian to mass-weighted Cartesian displacement coordinates and finally to the normal coordinates, Qlr by linear transformations with the result
D(QJ
=
Eb
if X z f = hrc5and where the constants yzi, X L 1 for the first 36 neutral elements have been given by Strand and Tietalo for the Hartree-Fock atomic fields. Values for y l i and Xii are also available for the ThomasFermi-Dirac field." The magnitude of the contribu~~~
~
~
(6) C.Tavard, private communication. (7) T. Iijima, Bull. Chem. SOC.Japan, 39, 843 (1966). (8)B . J. Laurenzi and D. D. Fitts, J . Chem. Phys., 43, 317 (1965). (9)For an explanation of the terms used in eq 8 the reader may consult the article, R. A. Bonham and L. S. Bartell, J . Chem. Phys., 31,702 (1959). (10) T.G.Strand and T. Tietz, Nuovo Cimento, B41, 89 (1966). (11) R. A. Bonham and T. G . Strand, J. Chem. Phys., 39, 2200 (1963).
MOLECULAR PROPERTIES AND ELECTRON DIFFRACTION INTENSITIES
tion from eq 29 is small compared to the experimentally dependent term' and its calculation depends only on a knowledge of the molecular structural parameters (rtj,lil) and the atomic fields ( y t t ,X l f ) . It should be argued a t this point that serious error may arise in the use of eq 27 if Hartree-Fock (HF) atomic wave functions are used in the calculation of the quantity I A ~ ~because ( ~ ) such wave functions neglect electron correlation. I n addition the interpretation may be affected by errors in the simple Born theory of scattering. Iijima" has shown how the correlation problem can be solved for those cases where the atomic correlation energies are known. If HF wave functions are employed, it should be noted that since the virial theorem holds, the total H F energy is given by
where I A ~ ~ (iss )calculated with HF wave functions. The exact wave function, on the other hand, can be = !VHF A, where A is a correction term written as to include correlation. The exact total energy is then given as
+
E
=
l/@HFILr/!PHF)
+ R , ( \ E ~ ~ I VI A ) + i / z ( ~VI A/)
859
Unfortunately the limit as a! + 0 does not exist so that it is not possible to obtain exact relations for the moments of r to the nth power by this means. On the other hand, for sufficiently small a! the approximate moment relation
N
+
dr{rk - ark+.. . . ) D n ( r )
- 2 5 2, n=l
1
dr{1x - ark+'. . . . } P ( r ) (34)
can be derived. Various difference formulas can be developed which may also prove useful. For instance if the left-hand side of eq 34 is denoted by c k , then it is possible to write moment relations such as
and finally
(31)
where the first term is the H F energy and the last two terms must be the correlation energy. From these -~ results it follows that HF atomic quantities may be employed in eq 27 if the sum Of the atomic 'Orrelation energies in the molecule is subtracted from the final result. Besides the foregoing relations presented by other authors it is possible to obtain additional relations by utilizing a generalization of eq 11 as well as some other transformations. For brevity vibrational averages have been omitted. For instance since r12n may be written
Similar relations for the other diffraction intensities can be derived in an identical manner. I n addition to eq 32-36 the interesting relations ('/zT')
1
N
dss2mED(s)= -2 n=l
Z,p(rn)
.v (1/27r2) J m 0
dss2aeED(s)= -2 n=l
Znp(rn)
+do)
(37)
+ $dr[p(r)I2 (38)
n m
(1/2~2)
the result for mED(s) is
J0
--
dss2ueXR(s)= $ dr[p(r)12
(40)
may be obtained by integrating eq 1-4 over d s directly and by making use of the properties of the Dirac -2
2
2,
n-1
l
drrne-*'Dn(r)
+
drr"e-"'P(r)
(33)
(12) T. Iijima, private communication.
Volume 71, Number 4
March 1867
860
R. A. BONHAM
6 function ( 4 ~ dss2jo(sr) c = 8r36(r)). These re-
sults are related to quantities used in nmr studies.I3 Of course the individual p(rn) are the quantities of greatest interest in nmr work, but the sum of the p(rn) values can serve as a check on theoretically computed values of electron densities at the nuclei and can also be used to supplement other data in order to determine p(r) from diffraction quantities. Besides the Fourier transform relation of eq 26 the Laplace transform of the intensities can be used as ass" = lim U-0
l
dssne-a' = lim -0
(-&- 1dse-a8
(41)
to obtain
o (r2
+
a2)
(42) Similar relations for the other diffraction intensities can be developed in the same manner. Here it appears that the case where k = -1 is finite in the limit as CY -+ 0 although the results for higher powers of (l/r) are probably singular in the limit. The result for k = - 1can be written as
sin (sr)/sr in eq 1-4 and by determining the expansion coefficients by least-squares analysis of the smallangle scattering data. For the total electron scattering this expansion leads to
mEDW =
[-($lz*)($lzf + 1) N
;{-2glz{l "(-2
N
120
i-1
drDi(r)r2
+
l
drP(r)r2}
drP(r)r4}. .
Z i c d r D t ( r ) r 4+ J m
+
.] (46)
0
from which it should be possible to obtain estimates of the sums of the lower order expectation values. At this point it appears that the only useful expectation values of a power of r which is missing in the present analysis is (l/r3) which is related to the nuclear paramagnetic shielding constant used in nmr work. Finally it is of interest to investigate the possibility of obtaining information about molecular electron densities in a different fashion than the one given by Tavard, Rouault, and R O U X . To ~ ~ do this it is convenient to expand the three-dimensional electron density p(r) and the three-dimensional electronelectron density p,(r) as P(r) =
2 5
n - 0 1 - -n
Pn(r>ynz(e,p)
(47)
and m
Pc(r) =
e
C 2 PnC(T)YnJ(e,p)
n - 0 I-. -n
(48)
where the YnJ(0,p) are the unnormalized spherical harmonics Pnr(cos0) cos Zp. The modified diffraction intensities can then be written as
As in the case of eq 33-36 it is possible to expand the exponential on the left of eq 42 with the result i
rn
m
where
and Dk+8
=
(-&)
k+z+l
0
drP(r) (r2 CY2)
+
Additional moments may be obtained by expanding The Journal of Phyeical Chemistry
_ _ _ _ _ ~ ~ ~~
~
(13) C. J. Jemeson and H. 8. Gutowsky, J . C h . Phys., 40, 1714 (1964).
MOLECULAR PROPERTIES AND ELECTRON DIFFRACTION INTENSITIES
and
Co(s) = 4a
km
drr2p2(r)jo(sr)
(55)
861
poC(r) on the left of eq 57 and 58 must then be reiterated to self-consistency. I n addition, the numerous expectation values discussed previously may be utilized as constraints along with the usual conservation of probability density utilized by Tavard and co-workers,2b that is
A procedure can now be set up to obtain the radial coefficients p,(r) directly from the intensities U T ~ ~ ( S ) or ueED(s), It is important to note at this point that in principle an approximate three-dimensional electron density can be determined from diffraction data for polyatomic molecules. The density is approximate because the terms D,(s)B,(s) are coupled by the vibrational average since the density quantities are functions of the nuclear positional coordinates. However these densities can be expanded in terms of normal coordinates about, the equilibrium structure as
The vibrational averages can then be carried out and the previous equations can be modified to include correction terms depending on the slope and curvature of the electron density a t the equilibrium nuclear positions. Even without this correction the uncertainty in the determination of the electron density should not be serious in the case of small molecules where bond distances are short and amplitudes of vibration are reasonably small. By use of the well-known Fourier-Bessel integral the density coefficients can be determined directly from the total scattered electron-diffraction intensity 89
and
Of course the quantities on the far right of eq 57 and 58 both depend on the unknown densities, so these terms must be evaluated using an initial guess as to the molecular density. The results obtained for p n ( r ) >
and Sdrp,(r) = 4a
&c
drr2poc(r) = M ( M
- 1)
(60)
Of course the utility of the above scheme will depend on the availability of accurate intensity data and a set of initial guesses for the pn(r) and poc(r) that are sufficiently close to the final result t o ensure convergence of the iterative procedure. It is difficult to judge at this early state of development of the experimental method whether or not sufficiently accurate data can be obtained. Nonetheless the foregoing points out the important fundamental difference between X-ray diffraction where threedimensional molecular charge density information is largely lost and the electron case where in principle the information is present. It is of interest to note that preliminary work on H2 and CH4 has yielded binding energies which are within 15% of the thermochemical ~ a l u e . ~ItJ ~is hoped that presently available experimental refinements coupled with a better understanding of scattering theory will yield sufficiently accurate data to make the above outlined approach feasible. As t o the problem of an accurate initial guess a possible solution might be to use a density based on a super position of spherical atom densities for the pn(r) terms. For the pOc(r)term a first approximation might be to let p o c ( r ) be given by ( M - 1). poO(r),where poO(r)is the zero-order approximation for po(r). The p n ( r ) can be approximated by taking a superposition of spherical atoms in the proper molecular geometry and then expanding the resultant molecular density in spherical harmonics. For the HartreeFock" or Thomas-Fermi-Dirac12 atomic densities, the general radial component pn(r) can be written as eq 61 where K is the number of terms in the potential field expansion, 7; and XI" are electron density parameters for the atomic field of the kth atom, and elk and cplk are orientation angles of the vector rlk t o the 2 axis r12. Equation 57 thus provides a convenient estimate with which to start the calculation of the molecular electron density. (14) T. Iijima, R.A. Bonham, C. Tavard, M. Roux, and M. Cornille, Bull. Chem. SOC.Japan, 38, 1758 (1965).
Volume 71,Numbm 4 March 1067
R. A. BONHAM
862
Finally, it should be emphasized that the ease in obtaining the various experimental intensity values as well as the accuracy with which they may be obtained may possibly be in the order mED(s)> ueED(s)> UTXR (s) > ueXR(s). Unfortunately, X-ray work of the type referred to here does not appear to be currently in vogue and it is difficult to determine what improvements might be made in X-ray techniques because of modern technological advances. At any rate it a p pears that because of the wealth of interesting information obtainable from such data that work in this area might fruitfully be reinstigated.
Acknowledgment. The author wishes to thank Professor Otto Bastiansen for his kind hospitality, Drs. Tavard, Iijima, and Strand for results in advance of publication, and Professor L. S. Bartell for his helpful comments and suggestions.
Tlre Joutml
of
P h y a h l Chemistry
