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Electric field gradients from generalized X-ray scattering factors
ELECTRIC
FIELD GRADIENTS
15 July 1977
CHEMICAL PHYSICS LETTERS
Volume 49, number 2
FROM GENERALIZED
X-RAY
SCATTERING
FACTORS
Robert F. STEWART Department of Chemistry. Carnegie-Mdlon Pittsburgh. Pennsylvania 15213. USA
University,
Reccivcd 19 April 1977
The electronic contributions to the clcctric field gradient for Nz are studied with the USCofgenerali7ed X-r&y scattcring factors. Vibrational effects are neglected. It is shown for this example that the rclcvant charge density features are well within the resolution of the X-ray diffraction eupcrimcnt. It is hoped that the discussion in this now will stimulate fruitful collaboration between X-ray diffractionists and NQR spectroscopists.
1. Introduction
There is some prospect that electric-field gradients may be reliably estimated from the charge density results of X-ray diffraction data. The electric-field gradient has the same units as the electron charge density function, p(r). The field gradient about a source point nucleus a does not depend on the charge density on nucleus a, but does depend on the a centered quadrupole component of the total charge density, p(r,). By contrast p(a) is very large at center a and cannot be reliably determined from X-ray diffraction data. The field gradient has an *g3 dependence, but the charge density p(r,) can fill up space as rz_ It is not clear just what regions of space from center a are dominant in the determination of the electric-field gradient for a molecule or crystal. In the present note the spatial aspects of the electric-field gradient for N, are explored with the use of generalized X-ray scattering factors.
molecular electric-field gradient at the nuclear center according to a general theorem reported in ref. [3] . The first three generalized X-ray scattering factors for the N pseudoatom in N7 are shown in fig. I_ For the monopole term a difference function, AfN 0 is displayed rather than fNso. In this case the scittering fac-
2. Generalized X-ray scattering factors for N, A [2/2] expansion of the X-ray molecular form factor for Nz has been determined in previous work [l] . The molecular form factor was based on an SCF wavefunction for Nz(~C~) at the experimental R, = 2.068 bohr [2] _The generalized X-ray scattering factors for the [212] expansion will necessarily give the
Fi$. 1. Gentxalizcd X-ray scattcring factors of the N pseudoatom for the [212] expansion of the SCI’ molecular form fxtor. The monopolc, AfN,o, includes subtraction ot the t4S) N atom scatterinp factor. 281
Volume 49, number 2 tor for a I-lartree-Fock tracted
out. The f$
(4S)
N atoln has been sub-
are
the Fourier-Bessel transforms of the several, radial multipole functions for the N pseudoatom,
(1) 0
In fig. 1 the dominant multipole scattering factor is the quadrupolar term,fN,-, , and, moreover most of its amplitude is within sin B/h = 1 .O a-l with the maxi~num in the function at sin O/1 = 0.4 A-1 _The essential features of the scattering factors shown in fig. I are well within the resolution of the X-ray diffraction experiment. It is informative to recast the results in fig. 1 in terms of direct density space and to study the contributions of each pseudoatom to the electric-field gradient.
3. Pseudoatom for N,
densities and the electric-field
gradient
The generalized X-ray scattering factors fN >k _are solutions to functional equations [3 ] so that they are determined numerically. In order to transfortn the fN,k to density functions it is convenient to fit these functions with orthogonal polynomials. We itnagine the pseudoatom multipole pa,[(ra) can be represented,
x L;‘+‘(rr,)
&OS
15 July 1977
CHEMICAL PHYSICS LETTERS
e,>,
(2)
where t,:‘+” (x) is a Laguerre polynomial of order 12 and degree 32+-2 and P,(cos ea) is the Legertdre polynomial for the fth multipole. The coefficients a,, and exponential parameter y can be detertnined by tnean square methods. It can be shown that the FourierBessel transform of the radially dependent part of p a.1(r a ) in (2) is,
Jacobi polynomial. Thk procedure is to fit thefN I(,!?) with the analytical form (3). Details of this methhd will be published later. In the least squares fitting of the molecular form factor, F,,,(S, R,) with generalized X-ray factors. minimization with respect to the Q centered pseudoatom quadrupole, is equivalent to solving for the Fourier-Bessel transform of the Q centered molecular quadrupole density function. This means in direct density space,
where, for the [212] expansion of N?, p&r,) is the radial quadrupole function of the N pseudoatom at center (I and &,&(rb) are the radial monopole, dipole and quadrupole functions for the pseudoatotn at center b. The a centered multipole expansion of the tnolecular density is defined by P,,&)
= (4n)-l,$
The electric-field 4,
= 2Z&
Pnloli(Qq.(cos
$)-
(5)
gradient at center a.
- 2 c[P mo,CW,Ccos $)rc3
dr,
(6)
where zb is the charge of nucleus b and R, is the intcrnuclear distance. When (5) and (4) are inserted into the second term of (6), the electronic charge contribution to the electric-field gradient from pseudoatom LIis, ~,,~(r~) = +(ra)pz(cos
e,)/4,
(7)
and from pseudoatom b, k&..)
= $ ii-l[ $O *_ JPb,&(‘b)pk(COS4J
x Iycos 0,) daa
1
$(cos
ea)_
(8)
A plot ofp,,2(Q> (7), as a contour map is shown in
2[-(2s/y)l
c b,,[ ff+3”+1’2)(r), (3) y - [(_s/++#+’ J* where b,, I =a,, 2”(n + 2Z+ 2)!/ [2(1+ n) + I] !! and t = [W/G)2 - 11/ [(2S/y)’ + I] _ In (3), Plop’ is a =
282
317
3
fig. 2. The maxima and minima are at a distance of 0.47 bohr (0.25 A) from the nucleus. The extension of this quadrupole pseudoatom density function is nominally a “valence” type density function_ [Recall that fN,2(S) in fig. 1 has a maximum at sin i3/X = 0.4 A-1 _] The contribution of pa,2(r;l) to the electric-
15 July 1977
CHEMICAL PHYSICS LETTERS
Volume 49, number 2
-. - _\ \ \ \
/
/
I
-b
-\
/ /
I
Fig. 2. Plot of the Q pscudoatom quadrupolc in Na [see eq. (7)]. Contours at + 0.01, f 0.03, + 0.05, + 0.07. 0.09,0.11, 0.13 and 0.15 e bohrw3_ (Dashed tines < 0.)
field gradient at a is given in table I. It comprises 55% of the total electronic contribution. A plot of the a centered quadrupole density from pseudoatom b [i.e. \~a,~(‘~) in (8)] is shown in fig. 3. Note that the maximum charge density is at nucleus b and comes from the monopole of pseudoatom b. The several contributions of pseudoatom b to the electric-field gradient are listed in table 1. The full radial dependence of the molecular charge density on the a centered electricfield gradient can be seen in a plot of
\
---_____---
\
Pig. 3. Plot of the a centcrcd quadrupolc due to pscudo.rtom b in N2 [see eq. (8) J _ Contours at -c 0.02, f 0.04, f 0.06, f. 0.08, + 0.10, + 0.15, -c 0.2, -’ O-3,0.5 c bohr-3. (&shed lines < 0.)
A graph of (9) is given in fig. 4. The area under the curve, when extended to ra + 00, is proportional to the field gradient from the electrons. It would appear that the monopole of pseudoatom b plays a dominant, but nearly indeterminant (in the X-ray sense) role in the electric-field gradient at center a. The isolated N atom (4S) at R, = 2.068 au from a, however, has a contribution of -1.23 1 au to the CI centered
electric-field
gradient.
Thus,
AfNao
on center b makes a very small contribution
in fig. 1
(+O.OOl
06,-
Table 1 Pseudoatom and nuclear contributions gradient in N2 (Re = 2.068 b&r)
to the electric-field
0.5 -
o-3-
Component
Electric-field (au) a) at 0
pa.2 @a)
-1.621
Db,o(‘b)
-1.230 0.001
APb,o(‘b) pb,t(rb)
-0.010
Pb,Z(‘b)
-0.087
0”
1.583
2z& qzz [es. (6)I a) 1 au = 3.2414
gradient
x 10”
-1.365 esu cm-j
= 9.7175
J m-* C-r.
02
04
06
06
10
r,,
12
14
16
18
2ob22
24
(bohd
Fig. 4. Plot of the radial dcpendcnce of the molecular charge density on the D centered electric-field gradient in N2 [see eq. (9)1-
283
Volume 49,
number 2
CHEhfICAL PHYSICS LE-i-fERS
15 July 1977
gradients in crystals. From my perspective, specimens of low 2 value (such as nitro-
au)_ In this context the ‘core” density of the b pseudoatom is insensitive to chemical bonding of N in Na _
nation of electric-field
(For other examples of pseudoatom monopoles in diatomic moIecuIes see figs. 1 and 8 in ref. [l] _)
gen-containing compounds) would be desirable. We have reported electric-field gradients, implied by a
4. Conclusion The analysis above has been carried out for the static charge density of the non-vibrating Nz (Ix:) molecule. It has been tacitly assumed that the SC?F wavefunction of Cade et al. [2] reveals the essentialIy correct. if not absolute features of the electric-field gradient for the N, molecule. At the static charge level it seems that the electric-field gradient for N, (as well as for other molecules) is undoubtedly within the
charge density analysis of X-ray diffraction
for l,I’-
azobiscarbamide [4], but I have no assurance if these results are even approximately correct. It would indeed be productive to bring NQR and X-ray diffraction experimentalists into a vigorous study of the same specimens. Perhaps a resolution similar to Kuchitsu and Cyvin’s work on spectroscopy and electron diffraction results [S] can be accomplished_
Acknowledgement
resolution of the coherent X-ray scattering experiment [(sin B/h),,, < 1.5 a-l ] . The Bragg structure fktors from accurate X-ray diffraction measurements are selected (by the von Lauc interference conditions) Fourier coefficients of the rfbratiorrali>r averaged charge density function in the crystallographic unit cell. Without a suitable vi-
The author is grateful to Mr. Joel Epstein for his computational efforts in converting the generalized X-ray scattering factors into charge density functions. This research has been supported by NSF Grant CHE
brational and/or Iibrational potential, only the eiectricfield gradient of the thernzal average of the charge den-
References
sity function can be determined_ On the other hand, in the nuclear quadrupole resonance experiment (as for 14N), the reduced result is fundamentally the thermal vibrational average of the electric-field gradient. It should be possible to bring the two experiments to a basis for comparison. The plea, therefore, is that NQR spectroscopists
[I ] J. Bentley and R.F. Stewart, J. Chcm. Phys. 63 (1975) 3794. [2] P.E. Cade and AC. Wahl, Atomic Data Nuclear Data Ta-
venture collaboration with X-ray crystallographers, interested in charge density analysis, on the detenni-
284
74-17592.
bles 13 (1974) 339. R-1‘. Stewart, J. Bentley and B. Goodman, J. Chem. Phys. 63 (197.5) 3786. [4 1 D.T. Cromer, A.C. Larson and RX. Stewart, J. Chem. [3]
Phys. 65 (1976) 336. [5 ] K. Kuchitsu and S-J. Cyvin, in: hlolccular structures and vibrations, cd. S.J. Cyvin (Elscvicr, Amsterdam, 1972) p_ 183.
FIELD GRADIENTS
15 July 1977
CHEMICAL PHYSICS LETTERS
Volume 49, number 2
FROM GENERALIZED
X-RAY
SCATTERING
FACTORS
Robert F. STEWART Department of Chemistry. Carnegie-Mdlon Pittsburgh. Pennsylvania 15213. USA
University,
Reccivcd 19 April 1977
The electronic contributions to the clcctric field gradient for Nz are studied with the USCofgenerali7ed X-r&y scattcring factors. Vibrational effects are neglected. It is shown for this example that the rclcvant charge density features are well within the resolution of the X-ray diffraction eupcrimcnt. It is hoped that the discussion in this now will stimulate fruitful collaboration between X-ray diffractionists and NQR spectroscopists.
1. Introduction
There is some prospect that electric-field gradients may be reliably estimated from the charge density results of X-ray diffraction data. The electric-field gradient has the same units as the electron charge density function, p(r). The field gradient about a source point nucleus a does not depend on the charge density on nucleus a, but does depend on the a centered quadrupole component of the total charge density, p(r,). By contrast p(a) is very large at center a and cannot be reliably determined from X-ray diffraction data. The field gradient has an *g3 dependence, but the charge density p(r,) can fill up space as rz_ It is not clear just what regions of space from center a are dominant in the determination of the electric-field gradient for a molecule or crystal. In the present note the spatial aspects of the electric-field gradient for N, are explored with the use of generalized X-ray scattering factors.
molecular electric-field gradient at the nuclear center according to a general theorem reported in ref. [3] . The first three generalized X-ray scattering factors for the N pseudoatom in N7 are shown in fig. I_ For the monopole term a difference function, AfN 0 is displayed rather than fNso. In this case the scittering fac-
2. Generalized X-ray scattering factors for N, A [2/2] expansion of the X-ray molecular form factor for Nz has been determined in previous work [l] . The molecular form factor was based on an SCF wavefunction for Nz(~C~) at the experimental R, = 2.068 bohr [2] _The generalized X-ray scattering factors for the [212] expansion will necessarily give the
Fi$. 1. Gentxalizcd X-ray scattcring factors of the N pseudoatom for the [212] expansion of the SCI’ molecular form fxtor. The monopolc, AfN,o, includes subtraction ot the t4S) N atom scatterinp factor. 281
Volume 49, number 2 tor for a I-lartree-Fock tracted
out. The f$
(4S)
N atoln has been sub-
are
the Fourier-Bessel transforms of the several, radial multipole functions for the N pseudoatom,
(1) 0
In fig. 1 the dominant multipole scattering factor is the quadrupolar term,fN,-, , and, moreover most of its amplitude is within sin B/h = 1 .O a-l with the maxi~num in the function at sin O/1 = 0.4 A-1 _The essential features of the scattering factors shown in fig. I are well within the resolution of the X-ray diffraction experiment. It is informative to recast the results in fig. 1 in terms of direct density space and to study the contributions of each pseudoatom to the electric-field gradient.
3. Pseudoatom for N,
densities and the electric-field
gradient
The generalized X-ray scattering factors fN >k _are solutions to functional equations [3 ] so that they are determined numerically. In order to transfortn the fN,k to density functions it is convenient to fit these functions with orthogonal polynomials. We itnagine the pseudoatom multipole pa,[(ra) can be represented,
x L;‘+‘(rr,)
&OS
15 July 1977
CHEMICAL PHYSICS LETTERS
e,>,
(2)
where t,:‘+” (x) is a Laguerre polynomial of order 12 and degree 32+-2 and P,(cos ea) is the Legertdre polynomial for the fth multipole. The coefficients a,, and exponential parameter y can be detertnined by tnean square methods. It can be shown that the FourierBessel transform of the radially dependent part of p a.1(r a ) in (2) is,
Jacobi polynomial. Thk procedure is to fit thefN I(,!?) with the analytical form (3). Details of this methhd will be published later. In the least squares fitting of the molecular form factor, F,,,(S, R,) with generalized X-ray factors. minimization with respect to the Q centered pseudoatom quadrupole, is equivalent to solving for the Fourier-Bessel transform of the Q centered molecular quadrupole density function. This means in direct density space,
where, for the [212] expansion of N?, p&r,) is the radial quadrupole function of the N pseudoatom at center (I and &,&(rb) are the radial monopole, dipole and quadrupole functions for the pseudoatotn at center b. The a centered multipole expansion of the tnolecular density is defined by P,,&)
= (4n)-l,$
The electric-field 4,
= 2Z&
Pnloli(Qq.(cos
$)-
(5)
gradient at center a.
- 2 c[P mo,CW,Ccos $)rc3
dr,
(6)
where zb is the charge of nucleus b and R, is the intcrnuclear distance. When (5) and (4) are inserted into the second term of (6), the electronic charge contribution to the electric-field gradient from pseudoatom LIis, ~,,~(r~) = +(ra)pz(cos
e,)/4,
(7)
and from pseudoatom b, k&..)
= $ ii-l[ $O *_ JPb,&(‘b)pk(COS4J
x Iycos 0,) daa
1
$(cos
ea)_
(8)
A plot ofp,,2(Q> (7), as a contour map is shown in
2[-(2s/y)l
c b,,[ ff+3”+1’2)(r), (3) y - [(_s/++#+’ J* where b,, I =a,, 2”(n + 2Z+ 2)!/ [2(1+ n) + I] !! and t = [W/G)2 - 11/ [(2S/y)’ + I] _ In (3), Plop’ is a =
282
317
3
fig. 2. The maxima and minima are at a distance of 0.47 bohr (0.25 A) from the nucleus. The extension of this quadrupole pseudoatom density function is nominally a “valence” type density function_ [Recall that fN,2(S) in fig. 1 has a maximum at sin i3/X = 0.4 A-1 _] The contribution of pa,2(r;l) to the electric-
15 July 1977
CHEMICAL PHYSICS LETTERS
Volume 49, number 2
-. - _\ \ \ \
/
/
I
-b
-\
/ /
I
Fig. 2. Plot of the Q pscudoatom quadrupolc in Na [see eq. (7)]. Contours at + 0.01, f 0.03, + 0.05, + 0.07. 0.09,0.11, 0.13 and 0.15 e bohrw3_ (Dashed tines < 0.)
field gradient at a is given in table I. It comprises 55% of the total electronic contribution. A plot of the a centered quadrupole density from pseudoatom b [i.e. \~a,~(‘~) in (8)] is shown in fig. 3. Note that the maximum charge density is at nucleus b and comes from the monopole of pseudoatom b. The several contributions of pseudoatom b to the electric-field gradient are listed in table 1. The full radial dependence of the molecular charge density on the a centered electricfield gradient can be seen in a plot of
\
---_____---
\
Pig. 3. Plot of the a centcrcd quadrupolc due to pscudo.rtom b in N2 [see eq. (8) J _ Contours at -c 0.02, f 0.04, f 0.06, f. 0.08, + 0.10, + 0.15, -c 0.2, -’ O-3,0.5 c bohr-3. (&shed lines < 0.)
A graph of (9) is given in fig. 4. The area under the curve, when extended to ra + 00, is proportional to the field gradient from the electrons. It would appear that the monopole of pseudoatom b plays a dominant, but nearly indeterminant (in the X-ray sense) role in the electric-field gradient at center a. The isolated N atom (4S) at R, = 2.068 au from a, however, has a contribution of -1.23 1 au to the CI centered
electric-field
gradient.
Thus,
AfNao
on center b makes a very small contribution
in fig. 1
(+O.OOl
06,-
Table 1 Pseudoatom and nuclear contributions gradient in N2 (Re = 2.068 b&r)
to the electric-field
0.5 -
o-3-
Component
Electric-field (au) a) at 0
pa.2 @a)
-1.621
Db,o(‘b)
-1.230 0.001
APb,o(‘b) pb,t(rb)
-0.010
Pb,Z(‘b)
-0.087
0”
1.583
2z& qzz [es. (6)I a) 1 au = 3.2414
gradient
x 10”
-1.365 esu cm-j
= 9.7175
J m-* C-r.
02
04
06
06
10
r,,
12
14
16
18
2ob22
24
(bohd
Fig. 4. Plot of the radial dcpendcnce of the molecular charge density on the D centered electric-field gradient in N2 [see eq. (9)1-
283
Volume 49,
number 2
CHEhfICAL PHYSICS LE-i-fERS
15 July 1977
gradients in crystals. From my perspective, specimens of low 2 value (such as nitro-
au)_ In this context the ‘core” density of the b pseudoatom is insensitive to chemical bonding of N in Na _
nation of electric-field
(For other examples of pseudoatom monopoles in diatomic moIecuIes see figs. 1 and 8 in ref. [l] _)
gen-containing compounds) would be desirable. We have reported electric-field gradients, implied by a
4. Conclusion The analysis above has been carried out for the static charge density of the non-vibrating Nz (Ix:) molecule. It has been tacitly assumed that the SC?F wavefunction of Cade et al. [2] reveals the essentialIy correct. if not absolute features of the electric-field gradient for the N, molecule. At the static charge level it seems that the electric-field gradient for N, (as well as for other molecules) is undoubtedly within the
charge density analysis of X-ray diffraction
for l,I’-
azobiscarbamide [4], but I have no assurance if these results are even approximately correct. It would indeed be productive to bring NQR and X-ray diffraction experimentalists into a vigorous study of the same specimens. Perhaps a resolution similar to Kuchitsu and Cyvin’s work on spectroscopy and electron diffraction results [S] can be accomplished_
Acknowledgement
resolution of the coherent X-ray scattering experiment [(sin B/h),,, < 1.5 a-l ] . The Bragg structure fktors from accurate X-ray diffraction measurements are selected (by the von Lauc interference conditions) Fourier coefficients of the rfbratiorrali>r averaged charge density function in the crystallographic unit cell. Without a suitable vi-
The author is grateful to Mr. Joel Epstein for his computational efforts in converting the generalized X-ray scattering factors into charge density functions. This research has been supported by NSF Grant CHE
brational and/or Iibrational potential, only the eiectricfield gradient of the thernzal average of the charge den-
References
sity function can be determined_ On the other hand, in the nuclear quadrupole resonance experiment (as for 14N), the reduced result is fundamentally the thermal vibrational average of the electric-field gradient. It should be possible to bring the two experiments to a basis for comparison. The plea, therefore, is that NQR spectroscopists
[I ] J. Bentley and R.F. Stewart, J. Chcm. Phys. 63 (1975) 3794. [2] P.E. Cade and AC. Wahl, Atomic Data Nuclear Data Ta-
venture collaboration with X-ray crystallographers, interested in charge density analysis, on the detenni-
284
74-17592.
bles 13 (1974) 339. R-1‘. Stewart, J. Bentley and B. Goodman, J. Chem. Phys. 63 (197.5) 3786. [4 1 D.T. Cromer, A.C. Larson and RX. Stewart, J. Chem. [3]
Phys. 65 (1976) 336. [5 ] K. Kuchitsu and S-J. Cyvin, in: hlolccular structures and vibrations, cd. S.J. Cyvin (Elscvicr, Amsterdam, 1972) p_ 183.