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On hybrid orbitals in momentum space
Z.B. Maksid and W.J. Orville-Thomas (Editors) Pauling's Legacy: Modern Modelling of the Chemical Bond
213
Theoretical and Computational Chemistry, Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.
O n H y b r i d O r b i t a l s in M o m e n t u m
Space
B. James Clark, Hartmut L. Schmider, and Vedene H. Smith, Jr. ~ * Department of Chemistry, Queen's University, Kingston, Ontario, Canada KTL 3N6 Hybrids constructed from hydrogenic eigenfunctions are examined in their momentumspace representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the toomentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 1. I N T R O D U C T I O N Among the countless concepts that Linus Pauling introduced from Quantum Mechanics into chemistry[I,2], and that became standard principles of the trade, there is the idea of "hybridization". In the framework of the valence-bond description of a system, it is useful to mix atomic orbitals of the same "n-quantum number", or of similar spatial extent, to construct directed, asymmetric atomic contributions. Although hybrids are not needed in an LCAO-MO description of the system, they have so much become part of the language of both organic and inorganic chemistry, that people will go out of their way to arrive at descriptions that are compatible with them. A much lesser known contribution of Pauling to the chemical knowledge, is his explicit expression for the momentum representation of the hydrogenic wave function [3]. Momentum space concepts are common among scattering physicists, some experimental chemists and a few theoreticians; however, they have not won over the bulk of chemists nearly as efficiently as the hybrid concept. The reason is that they are somewhat counter intuitive and molecular structure is expressed in a rather indirect and (in the truest sense of the word) convoluted manner. The two concepts have on occasion been brought together; Coulson and Duncanson[4] gave an explicit formula for sp-orbitals based on Slater type orbitals (STO's). Rozendaal and Baerends used hybrids to describe chemical bonding in a momentum representation [5], and more recently, Cooper considered the shape of sp hybrids in momentum space, and their impact on momentum densities [6]. We would like to have a closer look at them, in terms of their functional behavior, their nodal structure and their topology. We will do *The Natural Sciences and Engineering Research Council of Canada (NSERCC) supplied financial support for this work.
214
so with a focus on single-center contributions, in other words we will restrict ourselves to atoms, and will only touch upon molecular aspects. Our model system is a hydrogen-like ion, for two reasons: first it is simple, and fulfills the basic premises of energetic degeneracy of the angular contributions; secondly, it was introduced by Linus Pauling, and it is he whom we are honoring, after all. 2. F O U R I E R T R A N S F O R M S
OF P O S I T I O N - S P A C E H Y B R I D S
To obtain the momentum distribution ~rr due to a single hybrid orbital r it is necessary to perform a Dirac-Fourier transform. The square magnitude of the resulting momentum orbital r is the contribution of r to the momentum density;
~(ff) = (271-)-3/2 / ~)(r') e-iyyi'd~ -
(1)
Note, that the hybrid r is real. It may be written as a combination of an inversionsymmetric part Cs and an antisymmetric part Ca. The Fourier transform will map the former onto the real part of r and the latter on its imaginary part, both of which are symmetric themselves. As a result, the square-magnitude of r is inversion symmetric with respect to i7=0 (as momentum densities should be) [7]. Commonly (in position space), hybrid orbitals are written in terms of single-center linear combinations of basis functions that are themselves products of radial parts and real spherical harmonics. Let us consider
where we designate 0 as the polar angle, and r as the azimuthal one. Here, the following assumptions are made; the radial function Rz(r) is the same for all basis functions of the same "/quantum number", and its dependence on a "shell quantum number" n is of no consequence. The coefficients a~ describe the contribution of s, p, d, ... character to the hybrid, and the b~m govern the shape and orientation of that contribution. Stm are the real surface harmonics, defined in terms of the spherical harmonics (Ylm). Slr~ -- N •
{ Re(Yll.q) if m _> 0 Im(Yzlml)if m < 0
(3)
Here, N is a normalization factor, chosen such that f I Sire ]2 dgt = 1. The original justification for constructing hybrids was that the radial behavior (in position space) of the constituting functions is rather similar, and that they are energetically near-degenerate. These conditions are fulfilled exactly for the eigenfunctions of a one-particle Coulomb system, the hydrogen-like ions with nuclear charge Z. We will therefore illustrate a few concepts on those. The radial behavior of the hydrogenic eigenfunctions in position and momentum space is exponential and "Lorentzian", respectively, and their nodal structure depends on the associated Laguerre and Gegenbauer polynomials, respectively: RH(r)
--
2n(n + l)!
-~-l-~
(4)
215
N
R.(p)
=
(-i)~
p
2 Z ( n - l - 1)' l!(Z/n)t+2(4p) t+x p t+x 7r(n -4-l)! -((--'~n)2;p~)l+2 "'n-l-1
(p2/zjn 2) p -t- (Z/Tt) 2
(5)
The momentum-space expression was first given by Podolsky and Pauling [3] in 1929. Note, that for any real position function RH, the corresponding momentum radial function RH will be either purely real or purely imaginary, depending on whether the angular part of the orbital is even or odd (see also [8]). The factor ( - i ) l in Eqn.(5) has, e.g., the consequence that s-type and p-type functions do not "mix" in momentum space, which leads to hybrids that have a different nodal structure. 3. H Y B R I D S I N M O M E N T U M 3.1.
Hybrids
SPACE
of the spa-type
The most commonly encountered hybrids in organic chemistry are linear combinations of s and p-type orbitals. Depending on the linear coefficients of the real p-functions in the three Cartesian directions, the resulting set of hybrids will be oriented at different solid angles from each other. For example linear combinations of the form 1/2(s :t:px :t:py-t-pz) will yield hybrids with a tetrahedral angle among them. This basic geometry will be retained in momentum space, since the Fourier-transform (1) is a direction-preserving, unitary transformation. However, the fact that even and odd contributions to the positionspace hybrid are transformed separately (via cosine and sine transforms) into real and imaginary parts of the momentum-hybrid, means that the resulting densities (i.e., the square magnitudes) are inversion symmetric with respect to the origin, which is obviously not the case in position space. Therefore, an arrangement of orbitals in a point group G in position space, will lead to an arrangement in the point group G' = G x Ci in momentum space; this is the direct product of G with the inversion-symmetric group (see [7] and [9]). For the following considerations, the orientations of the hybrids are not relevant, and we therefore mix only pz orbitals with s-functions, resulting in an orbital that points along the polar axis z in both spaces. This takes the following form for the n = 2 shell:
_ ~
+
-
[2-
+
co
0]
=
,
(6)
S i~r(1 + a) r
= -~
(r
In position space, r 1 + v/-acos 0 r - - 2 Z(a cos 2 0 - 1)"
+ x/~r
- 16 Z s/2 [4p 2 - Z 2 - 4 ix/-dZpcosO] 7 r ~ + a ) ( Z 2 + 4p2) 3
(7)
is real and its nodal surface is defined by
(8)
In contrast, the real part of the momentum hybrid ~,~ has a spherical node (at p -
~ r ~ } _ Z/2), whereas the imaginary part has a planar one (at 0 - 7c/2). This means
216 ~a
that the subspace, where r - 0, is one-dimensional. It is, in fact, a circle of radius p = Z / 2 in the xy-plane 2, with center at p = 0. Coulson and Duncanson [4] obtained an analytical expression for a momentum hybrid of a C-atom, constructed from STO's that shows similar structure. Later, Cooper and Loades gave contour plots of several similar hybrids in momentum space [6]. However, neither one did comment on this interesting feature. It is widely known that total momentum densities for atoms are not always monotonically decreasing [10]. In fact the degree of non-monotonicity is dependent on the degree of p-population in an atom. This fact is visible as well in the shape of s p a hybrids in momentum space.
Figure 1. Surface plot of the orbital densities for sp a hybrids in momentum space. The hybrids are based on the hydrogenic wave functions. The three plots pertain to a = 1, 2 and 3, respectively. The hybrids point in the z-direction. A section through the density in the xz-plane is displayed.
In Fig.(1), we display the momentum density contributions of commonly encountered hybrid orbitals, obtained from hydrogenic eigenfunctions with Z = 1. The figure shows surface plots of the densities for s p a in the xz-plane for a = 1, 2 and 3. It may be seen that, while the s p hybrid exhibits a maximum at p = 0, greater p-contributions flatten this maximum out, leading to a plateau for s p 2, and finally a saddle point for s p 3. All of these densities feature two points in the xz-plane where the density vanishes exactly. They are situated on the x-axis, as sections along that axis demonstrate clearly. We show those in Fig.(2). Independently of the mixing coefficient a, those "nodal points" occur at x = -1-1/2 on each equatorial axis. They are the intersection of the aforementioned nodal circle with the displayed plane. To further assess the extremal structure of the densities corresponding to our hybrids, we also display the curvature of the density along the polar axis in Fig.(3). The analytical expression for these curves is 2Throughout the article, we use the notation x, y, z for the Cartesian components of the momentum vector p, and 0, r for its angular spherical coordinates. This is done to avoid excessive subscripts, and confusion with p-orbitals.
217
027r~p(ff) I
8192(a
-
2)
p=o = Z57c2( a + 1)
OP2z
(9)
Therefore, clearly the sp2-hybrid is the limiting case, for which the transition from maximum to saddle-point occurs, independently of the nuclear charge Z. We note that in the paper of Cooper and Loades [6], this distinction was not found for the momentum hybrids. Using Gaussian orbitals, with exponents tailored for the carbon atom, the origin is a saddle point for all three hybrids. For the hybrids that we have constructed, maxima will occur for any a > 2 at
Z Pmax -- ~-
(6 - 10a + 2x/i " 26a + 25a 2) 1/2
(10)
As we should expect, increasing the amount of p character, by increasing the value of a, will shift the maxima of the momentum distribution, to higher values of p. This fact well illustrates Epstein and Tanner's H y b r i d Orbital Principle, which states that "increased p character in an s - p type hybrid orbital results in increased density at high momentum". [11] In addition, one should contrast the appearance of the plots from Fig.(1) with our more familiar position space representation of these sp a hybrids, as shown in Fig.(4). There, we always see two distinct maxima [6], the sharper of which is located right at the origin.
3.2. Hybrids involving d-orbitals For most of organic chemistry the description in terms of sp a hybrids is sufficient for a qualitative picture. However, if the coordination numbers involved are greater than 4, as is the case for the majority of compounds involving transition metals, d-hybridization has to be taken into account. Since the m-quantum number of a d-function influences not
Figure 2. Sections through the orbital momentum densities displayed in Fig.(1) along the x-axis with z = 0, i.e. perpendicular to the main axis of the hybrid. The three plots pertain to a = 1, a = 2 and a = 3, respectively. Note that each density vanishes at two points on the x-axis.
o-1I
o-1I
0.08
01I
0.08
0.08
0"02 I -1
-0.5
oo '
-
-
.
oo
oxs
1
~
5
'oo
218
Figure 3. Second derivative of the orbital momentum density, 027r~p(ff)/Op~, in the z-direction, along the polar axis. Note that for a = 1 (left plot), this quantity is negative around p = 0, for a = 2 (middle plot) it is exactly zero, indicating a plateau, and for a = 3 (right plot) it is positive, denoting a saddle point.
ioo
-
2oo~
Figure 4. The sp, sp 2, and sp 3 hybrid density functions in the xz-plane of position space. As is often the case, orbitals that are quite different from one another in momentum space, can appear very similar in the corresponding position space representation.
only the orientation, but also the shape of the constituting atomic orbitals, we have to distinguish several cases. In the following, we focus on two of them; the first are hybrids that are directed along the z-axis, such as the ones in an octahedral complex, which have the general form 3sp~dbz2. The second lies in the xy-plane, and is exemplary of a hybrid that would be used to describe bonding in square-planar compounds. These hybrids are of the form 2 spxax2_y~ a-b . Other combinations are possible, but will in general show similar structural features.
219 The first class takes the following form (for the n = 3 shell) in momentum space:
~3sp~d2-a,b=
x/'l +la + b (~3s (#) + k/cd~3p~ (#) + k / ~ 3 d 2 (#))
18 3x/3--Z~ (81v/-2p4
--
6Z2p 2 [5~/-2 + 4x/b (3 cos20 - 1 ) ] + v/2Z 4)
(7rV/1 + a -t-b) (Z 2 + 9p2) 4 i 432pcosOv/Z7a
+
(Z 2 - 9p 2)
(~rv/1 -'~ a --}-b) (Z 2 nt- 9p2) 4 "
(11)
In the above equation, the hybrid is clearly broken down into a real part (second line), and an imaginary part (third line). We have found it convenient to analyze these two parts of the hybrid separately because of the earlier mentioned property that the real part of a hybrid will not "mix" with the imaginary part when computing expectation values and densities. The real part of Eqn. (11) arises from the mixing of s- and d-contributions. It has roots on either side of the xy-plane, on two closed C-rotation-symmetric surfaces, that are
Figure 5 9 Nodal surfaces of a s~3d ~'z 2z2 hybrid orbital with Z = 1 in the momentum-space representation. The left-hand plot contains two surfaces. One is the spherical node of the imaginary part. The second more complex surface consists of two closed and flattened spheres. These are the nodal surfaces belonging to the real part of the hybrid and are aligned along the z-axis. The intersection of the two types of nodes are two circles around the z-axis. The right-hand plot displays a cut through the xz-plane. Note that the (polar) z-axis is the horizontal axis in this plot. To avoid confusion, the nodal planes of the imaginary part are not displayed in either graph.
1IX f .. ...............~.5
"'\ z /,," ....................
-0.5
220 the solutions of a quadratic equation in p2. Figure (5) shows these surfaces (left plot), as well as a section through the zz-plane (right plot). The central sphere in the left plot and the circle in the right plot are nodes of the imaginary part of the hybrid. This imaginary part (third line of E q n . ( l l ) has nodal surfaces which consist of the xy plane (0 = 7r/2, not shown in the plots), and a sphere centered at p = 0 with radius Z/3. As ~a b a result, the density (i.e. the square magnitude of Cs;d,z), vanishes on a pair of circles at pz = =i=z [3 ( 1 - ~ r 1,2 with radius gz [ 2 / 3 + x/~/6v/~l U2 These circles are the intersection of the roots o t the real and the imaginary parts.-This nodal behavior is in clear contrast to the one in position space, where nodal surfaces of a rather complex shape are observed.
Figure 6. Surface plots of the momentum density corresponding to two different hybrids of the 3 b form sp~dz2. For the left plot, b = 2 (as in octahedral hybrids), for the right one, b = 5 (maximum bond strength). The surface plots are sections through the xz plane. Note that they both exhibit four points where the density vanishes exactly.
Fig.(6) shows the momentum densities corresponding to two different hybrids of the type. The first one is a sp3d 2 orbital as is encountered in octahedral complexes, the second one is a "maximum-bond hybrid" sp3d 5. The basic features are rather similar, although somewhat more strongly pronounced in the one with the greater d-component. The plots show a section through the densities in the zz-plane. Particularly for the sp3d 5 hybrid, a set of 4 "holes" parallel to the x-axis can be observed, arising from the aforementioned circular nodes. To show these "holes" more clearly, we have plotted in Fig.(7) circular sections through the densities, passing through the xz-plane with radius 1/3. The graphs show the momentum density as a function of the polar angle 0 in units of 7r. Note that the density is maximal inthe z-direction (0 = tTr;t = 0, 1,2), and in x-direction (0 = tTr;t = 1/2, 3/2), s -pz%~ a ..tb
221 Figure 7. Circular sections through the momentum densities displayed in Fig.(6). The plots display the value of the density along a circle of radius 1/3 in the xz-plane, as a function of the polar angle 0 -- tr. The nodal points are clearly visible.
, / 0
0.5
1 t
;0
I 5
k,o .
as would be expected and is also observed in position space. However, at angles of 00 and
[
7 r - 00 with 00 = a r c c o s ( ~ 2 ), the density goes exactly to zero, since this is the angle under which the real and imaginary nodes intersect. This angle is not dependent on Z, but depends weakly on b. It varies between 7r/2 for b = 1/2 and ~ / 2 - a r c s i n ( 1 / x / ~ ) for b ~ e~. The second class of spd-type hybrids that we will treat here are situated in the equatorial plane, i.e. at 0 = ~-/2. They consist of a linear combination of s, px and d~2_y~, orbitals, and have proven useful in describing the bonding in square-planar complexes. Their form (in momentum space) is:
Csp~d~2_y2-ab' (P) = X/'i +la + b (~3s(~[~) -3t-v/a~3p~ (]~) -+- v/b~3dz2-y2 (g)) 18Z 5/2 (x/~ (81p 4 + 2 4) - 622p 2 (12~/bcos(2r
5v/-6))
(~x/1 + a + b) (Z 2 + 9p2) 4 i 432 px/-Z-~ sin 0 cos r (Z 2 - 9p 2)
+
(~x/'l + a + b ) (Z 2 +
9p2) 4
(12)
Note that the imaginary part is essentially the same as in Eqn.(11), but the real part differs in shape. However, the consequence is a qualitatively different shape of both the nodal surfaces for the real part, and the nodal curves for the density. Figs.(8-10) show various aspects of the resulting density contribution for a 3sp~dx2_y2
222 Figure 8. Plots of the nodal surfaces in a sp~dx2_y2 hybrid orbital (Z = 1) in the momentum representation. The left plot shows the surface due to the real part (i.e., s and d contributions) only, whereas the right one combines it with the planar and spherical nodal surfaces characteristic of the imaginary (i.e., p-) component.
orbital, as this is the one used to describe square-planar complexes. Since the real part of Eqn.(12) is a linear combination of spherically symmetric s-contributions (with 2 spherical nodes) and "Rosetta-shaped" d-functions the resulting nodal surface is somewhat "donutshaped" around the y-direction. It is displayed in the left plot of Fig.(8). On the right-hand side we combined this surface with the planar and the spherical node of the imaginary part, which are due to the hybrids' p-contributions. Cuts through the xy-, xz- and yz-planes may serve to clarify the topology further (see Fig.(9)). The features in the two planes that are cut by the donut (xy and yz), are rather similar to the ones encountered earlier for the axial hybrids s P-a'b z t t z2. However, in the plane of the ring (xz) it differs considerably, and shows no intersections. All curves are rather contained within each other. Note also that the yz-plane is in itself a nodal plane of the imaginary part, and that therefore the cuts arising from the real part are nodes of the density. The intersections of the circle and the "outer" curves in the right plot are in fact intersections of 3 nodal surfaces, 2 from the imaginary and one from the real part. The lower right plot of Fig.(9) shows the nodes of the density (i.e., the intersections in 3Dspace). The two closed curves in the y z-plane arise from intersections of the donut-shaped node of the real part with the planar node of the imaginary one, whereas the other curves are intersections of the spherical node of the imaginary part with the donut, close to its "hole". The explicit expressions for the curves displayed in the lower right plot can be derived. They are: Z
p--~(V/4-x/~sin20•
;r
•
(13)
223 Figure 9. Traces of the nodal surfaces in the xy-, xz- and yz- planes for the 8p2dx2_y2 hybrid. It can be seen that while the basic features in two of the planes (xy and yz) are rather similar to the one observed in 8-ad pz b.2 type hybrids, the situation in the xz-plane is completely different. The lower right plot shows the nodal lines that are the intersections of various surfaces displayed in Fig.(8).
9f /'/l(,~ .... ',, -~ t '~ ;~176176
\.._ .....~
x
........./
-0"5t -1
1Y z
0.5
/
(o2
z \ . . . . . .
4
x
O.
-0.
t,.,
-0"5 I -1
r
2
2arcc~
V~sin2 0
;p=
It is rather surprising that such simple linear combinations will produce such complicated topologies in momentum space. Fig.(10) shows a surface plot of a section of the momentum density in the xy-plane, where the density is accumulated (left). The seemingly monotonous distribution exhibits on closer inspection a good deal of fine structure: first, there are the aforementioned "holes" in the vicinity of the nodal lines; secondly, the apparent maximum reveals itself on an enhanced scale (right plot), to be a saddle point that is minimal in the x-direction.
224 2 Figure 10 9 Surface plot (left) of the momentum density corresponding to a spxdz2_y2 hybrid orbital. The section displayed lies in the my-plane. Although the overall features seem to be rather weak, a complicated nodal structure is observed, and the origin is a saddle point (right). The right plot shows the momentum density on the x-axis.
49.25
-0.03
-0.02
-0. Ol
0
0. Ol X 0 . 0 2
0.03
For all momentum densities, the origin of momentum space is necessarily a critical point. This arises from the inversion center and the requirement of continuity. However, the topology at that point may vary considerably for the hybrids considered here. We pointed out already in the previous section that the spa-hybrids do not always exhibit a pair of off-center maxima. For hybrids containing d-functions, the picture is further complicated 9 To obtain a clear idea, it is best to obtain the diagonal elements of the Hessian matrix of the density Hij = 02rr~(p-)/OpiOpj in Cartesian coordinates 9 From their sign it may be inferred what type of critical point is observed. For the sp~dbz2-type hybrids, this yields
(Oq27r~bOq27rrOq271-~b)(2~/~ 11 2V/~ 0p
'op--7'0p
-
V27rr
=
-
r ( 8 a - 33)
'
-
l18a '
4V/~ -
11)
(15)
-
(16)
where r = 46656/Z57r2(1 + a + b) is a common pre-factor. The eigenvalue structure in Eqn.(15) indicates that p = 0 is a minimum in z-direction, whenever b < 2a 2
lla 121 --~+--~
and
a > 11/8,
(17)
These conditions are independent of Z, since the latter determines only the spatial extent, but not the basic topology. From Eqn.(15) we also infer that d-contributions tend to flatten out the minimum at the origin (see also Fig.(6)) by moving the maxima towards higher momenta, whereas
225 Figure 11. Map for the topology of the momentum density at the origin of momentum space p 0, for hybrids containing d-functions. The left plot is for functions of the Spzadbz2-type, the right one for the Spzadb~2_y2-type. The axes are the mixing coefficients a and b, respectively, and the lines drawn separate regions of different topology. =
35
60I
30'
5O
:o
ring
~~t
/
i0_
25' saddle
20 b 15
rlng
minimum
.
I0
l
5 maximum ,
O'
.0,0
~~ '
'
i
,
J ---~, .... 2
a
saddle ~
3
,
~1 . . . .
p-contributions sharpen the minimum, although not deepening it. In the x-direction (or 121 more generally, perpendicular to the polar axis), p = 0 is a maximum for b < - U , i.e. in all cases of practical interest. The left plot in Fig.(ll) shows the regions of different possible topologies at p - 0 in the (a, b) plane. Note that for "reasonable" hybrids, a < 3, and the origin in momentum space is therefore either a saddle point or a maximum. Rings or minima would only occur with very low relative s-contributions. Note that the Laplacian (16) (as the sum of second derivatives) at the origin does not depend on the d-contributions. They merely serve to "rearrange" the density, whereas p-components will make the Laplacian less and less negative. For the other type of d-containing hybrids, the spxdx2_v2 hybrids, we have
op~'
Op~'
Op~ v~r
~-
'
~ (Sa - 33)
11' -11/ (19)
where T is the same as in Eqn.(15). This means, the density is always a maximum in the z-direction at p = 0 for this type of hybrid. In the x-direction (which is the main axis of the hybrid), it is minimal if b<~8a2
22a3 ~ 12124 and
a > 11/8,
(20)
and in the y-direction it is maximal as long as b < 121/24 ~ 5.0417, i.e. up to and including spxads~2_y2 hybrids The Laplacian is the same as for the previous type, which is
226 a trivial consequence of the fact that it depends (for p = 0) only on the radial part of the constituting orbitals. Note that our example (sp~dx2_y2) has a saddle point of the momentum density at the origin. This fact can not be seen at the resolution of Fig.(10), since the saddle is extremely shallow (the second derivative is "only" 11664(5- 2v/g)/Tr 2 ~ 911.4. The right-hand plot in Fig.(11) shows the relevant section in the (a,b) plane with the proper designations for the type of critical point occurring at the origin. Note that a cage-structure is impossible since the hybrid is always concentrated in the xy-plane. Table 1 Moments of the sp a hybrids in both coordinate and momentum spaces. (r ~) n--
2
1 (3+a)Z 2 12 l+a
n--
1
1Z
128 a+2 15 ZTr(l+a)
6+5a
16 Z(3+4a) 45 7r(l+a) ,,,
4 39+7a
3 Z2(l+a)
n--O n--1
z(i+~)
n--2
6z2(t+~)
n--3
30 Z3(l+a)
n--4
240 Z4(l+a)
5a+7
1 2 ~Z
11+7a
12+7a
15 7r(l+a)
1_z4(37+7~) 48
l+a
4. M o m e n t s of the H y b r i d Orbitals
It is a simple matter to derive expressions for the moments of the hybrid orbital densities. In momentum space, the expressions will take the form, f pnrr~(g)dg = f ({a})Z n
(21)
where f ({a}) is a function of the mixing coefficients ( one mixing coefficient for the 8p a hybrids, two mixing coefficients for the sp~d b hybrids, etc.). The analogous expression in position space has an inverse dependence on the atomic charge, Z.
/ r'~pr
" - g ({a}) Z-~
(22)
where g ({a}) is also a function of the mixing coefficients, although different from the f function in Eqn. (21).
227
Complete expressions for the hybrid charge density moments, in both position and momentum space, are given in Tables 1 and 2. The former table contains results for the sp a hybrids while the latter gives results for the spad b hybrids. Table 2 Moments of the spa d b hybrids in both coordinate and momentum spaces.
(p')
(<) 2
n -- --2
n-
405
(15+3b+5a)Z 2 1+aTb
9 105+9b+25a
5 Z2(l+a+b) 16 200a+128b+335 175 ZTr(l+a+b)
1
n-0 8 Z(135+160aq-192b)
1 27+45a+21b
n-1
i575
2 Z(l+a+b) 20a+14b+23
Ir(1-t-a+b)
9 Z2(l+a+b)
1Z 2
n-3
81 70a+42b+85
16 Z3(200a+128b+335) 14175 rr(l+aTb)
n-4
405 238a+126b+303
n-
2
2 Z3(l+a+b) 2
Z4(l+a+b)
1 Z4(lO5-t-9b+25a) 405
l+a+b
Notice that the (r -1) and the (p2) moments have no dependence on the mixing parameters. They are simple functions of Z. This is a feature unique to the case where the hybrid orbitals are constructed from degenerate atomic orbitals. Since hydrogenic functions with the same n quantum number fulfill this condition, the "energy" moments, (r -1) and (p2}, will not depend on how the hybrid orbitals are mixed. The moments in Table 2 also do not depend on what types of d functions are used to construct the hybrids. In other words, the moments do not depend on the m quantum number. For example, a hybrid which uses only dz~ orbitals, will have the same moments as a hybrid which uses only dx2_y2 orbitals. This is, perhaps, an obvious point as the moments we are discussing are radial moments; angular characteristics in the hybrid density are spherically averaged during the integration over the solid angle. To see this more clearly, note the form of the integral (pn) for the case of an sp a hybrid.
(pn) --1
1 +a S (R2,o(P)Yo,o+ V/-aR2,I(P)YI,m)9pn (/~2,0(P)yo, 0 -t- %lraI~2,l(P)Yl,m)dp
(23)
The orthogonality of the spherical harmonics, Y/,m, insures that there will be no mixing of radial terms, Rn,t, in the resulting integral. Since the spherical harmonics are also normalized to unity, simplification of Eqn.(23) will produce the following functional form.
(P'~)spo h y b r i d
--"
1 +1 a
/
+ aR ,l (p)*p'R%I(;)] dp
(24)
228
(25)
l+a
Therefore, the hybrid density moment is simply a properly weighted average of each individual orbital moment. The orthogonality of the spherical harmonics has insured that different radial functions do not mix together. An analogous expression holds for the radial moments of the sp~d b hybrids.
1
(pn ) spadb hybrid -- 1 ~- a Jr b ( (pn ) s ~
-~- a (pn ) p ~
-~- b(pn ) d ~
)
(26)
5. C o n c l u s i o n In 1929, Linus Pauling, together with Boris Podolosky, became the first person to publish the m o m e n t u m representation of the eigenfunctions of a single-particle Coulombic Hamiltonian. Although he did not publish any more work on momentum space concepts, he is nonetheless a pioneer in the field of Momentum Space Quantum Chemistry. In this work, we brought together two of Pauling's contributions to Chemistry. The first, as mentioned above, is his pioneering work in momentum space quantum mechanics. The second is the concept of hybrid orbitals, originally used to understand the strengths and directional characteristics of covalent bonds. Accordingly, we have combined these two areas by looking at the form of some of the most useful hybrid orbitals, in momentum space. Among the more interesting qualities of momentum space hybrids, is the lack of strong directional asymmetry, this being one of the most noticeable characteristics of position space hybrids. Indeed, it is this directional asymmetry which has made hybrid orbitals so useful for describing directional bonds. In momentum space, however, the hybrids are inversion-symmetric and this is shown to have a profound effect on the nature of these orbitals. Another difference is the nodal structure of these atomic contributions to the total density. The hybrid orbitals as we know them, in position space, exhibit nodal surfaces, i.e. two-dimensional subspaces on which the density vanishes. This dimensionality is reduced in momentum space. Here, the nodes are invariably one dimensional, i.e. curves that are formed by the intersection of real and imaginary nodal planes. Finally, (for atoms), the momentum densities corresponding to hybrid orbitals exhibit a few basic extremal features close to the origin. These depend on the weight that is given to s, p and d contributions, and they determine the basic "look" of the density. Outwardly, momentum-space hybrids share one feature with a related experimental quantity, the Compton profile: they all look alike. On closer inspection, however, there are a variety of complex features, mainly arising from the nodal structure of the orbitals. Apart from the obvious use of hybrids in position space for the description of bond situations, there is another feature that has always captured the interest of scientists and laymen: their intricate structure. This feature is less apparent in momentum space, but it is still present. If nothing else, its enjoyment makes a close look at these entities worthwhile.
229 6. Acknowledgments We are grateful to Dr. Jiahu Wang and Minhhuy H6 for fruitful discussions, and to Dr. Zelek Herman for making the Pauling bibliography available to us. REFERENCES 1. Z. S. Herman. Some early (and lasting) contributions of Linus Pauling to quantum mechanics and statistical mechanics. In Molecules in Natural Science and Medicine: An Encomium for Linus Pauling, Z. Maksid and M. Eckert-Maksid, Eds. Ellis Horwood Limited, West Sussex, England, 1991, pp. 179-200. 2. Linus Pauling. The Nature of the Chemical Bond. Cornell University Press, Ithaca, New York, 1948. 3. B. Podolosky and L. Pauling, Phys. Rev. 36 (1929) 109. 4. C.A. Coulson and W.E. Duncanson, Proc. Cambr. Phil. Soc. 37 (1941) 67. 5. A. Rozendaal and E.J. Baerends, Chem. Phys. 95 (1985) 57. 6. D.L. Cooper and S. D. Loades, J. Mol. Struct. 229 (1991) 189. 7. P. Kaijser and V.H. Smith Jr. In Methods and structure in quantum science, J. Calais, O. Goscinski, J. Linderberg, and Y. Ohrn, Eds. Plenum Press, New York, 1976, pp. 417-426. 8. P. Kaijser and V.H. Smith Jr. Evaluation of momentum distributions and Compton profiles for atomic and molecular systems. In Advances in Quantum Chemistry, vol. 10. Academic Press, New York, 1977, pp. 37-76. 9. S.R. Gadre, A.C. Limaye and S.A. Kulkarni, J. Chem. Phys. 94 (1991) 8040. 10. W.M. Westgate, A.M. Simas and V.H. Smith Jr., J. Chem. Phys. 83 (1985) 4054. 11. I.R. Epstein and A.C. Tanner. In Cornpton Scattering, B. Williams, Ed. McGraw-Hill, New York, 1977, pp. 209-233.
213
Theoretical and Computational Chemistry, Vol. 6 9 1999 Elsevier Science B.V. All rights reserved.
O n H y b r i d O r b i t a l s in M o m e n t u m
Space
B. James Clark, Hartmut L. Schmider, and Vedene H. Smith, Jr. ~ * Department of Chemistry, Queen's University, Kingston, Ontario, Canada KTL 3N6 Hybrids constructed from hydrogenic eigenfunctions are examined in their momentumspace representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the toomentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 1. I N T R O D U C T I O N Among the countless concepts that Linus Pauling introduced from Quantum Mechanics into chemistry[I,2], and that became standard principles of the trade, there is the idea of "hybridization". In the framework of the valence-bond description of a system, it is useful to mix atomic orbitals of the same "n-quantum number", or of similar spatial extent, to construct directed, asymmetric atomic contributions. Although hybrids are not needed in an LCAO-MO description of the system, they have so much become part of the language of both organic and inorganic chemistry, that people will go out of their way to arrive at descriptions that are compatible with them. A much lesser known contribution of Pauling to the chemical knowledge, is his explicit expression for the momentum representation of the hydrogenic wave function [3]. Momentum space concepts are common among scattering physicists, some experimental chemists and a few theoreticians; however, they have not won over the bulk of chemists nearly as efficiently as the hybrid concept. The reason is that they are somewhat counter intuitive and molecular structure is expressed in a rather indirect and (in the truest sense of the word) convoluted manner. The two concepts have on occasion been brought together; Coulson and Duncanson[4] gave an explicit formula for sp-orbitals based on Slater type orbitals (STO's). Rozendaal and Baerends used hybrids to describe chemical bonding in a momentum representation [5], and more recently, Cooper considered the shape of sp hybrids in momentum space, and their impact on momentum densities [6]. We would like to have a closer look at them, in terms of their functional behavior, their nodal structure and their topology. We will do *The Natural Sciences and Engineering Research Council of Canada (NSERCC) supplied financial support for this work.
214
so with a focus on single-center contributions, in other words we will restrict ourselves to atoms, and will only touch upon molecular aspects. Our model system is a hydrogen-like ion, for two reasons: first it is simple, and fulfills the basic premises of energetic degeneracy of the angular contributions; secondly, it was introduced by Linus Pauling, and it is he whom we are honoring, after all. 2. F O U R I E R T R A N S F O R M S
OF P O S I T I O N - S P A C E H Y B R I D S
To obtain the momentum distribution ~rr due to a single hybrid orbital r it is necessary to perform a Dirac-Fourier transform. The square magnitude of the resulting momentum orbital r is the contribution of r to the momentum density;
~(ff) = (271-)-3/2 / ~)(r') e-iyyi'd~ -
(1)
Note, that the hybrid r is real. It may be written as a combination of an inversionsymmetric part Cs and an antisymmetric part Ca. The Fourier transform will map the former onto the real part of r and the latter on its imaginary part, both of which are symmetric themselves. As a result, the square-magnitude of r is inversion symmetric with respect to i7=0 (as momentum densities should be) [7]. Commonly (in position space), hybrid orbitals are written in terms of single-center linear combinations of basis functions that are themselves products of radial parts and real spherical harmonics. Let us consider
where we designate 0 as the polar angle, and r as the azimuthal one. Here, the following assumptions are made; the radial function Rz(r) is the same for all basis functions of the same "/quantum number", and its dependence on a "shell quantum number" n is of no consequence. The coefficients a~ describe the contribution of s, p, d, ... character to the hybrid, and the b~m govern the shape and orientation of that contribution. Stm are the real surface harmonics, defined in terms of the spherical harmonics (Ylm). Slr~ -- N •
{ Re(Yll.q) if m _> 0 Im(Yzlml)if m < 0
(3)
Here, N is a normalization factor, chosen such that f I Sire ]2 dgt = 1. The original justification for constructing hybrids was that the radial behavior (in position space) of the constituting functions is rather similar, and that they are energetically near-degenerate. These conditions are fulfilled exactly for the eigenfunctions of a one-particle Coulomb system, the hydrogen-like ions with nuclear charge Z. We will therefore illustrate a few concepts on those. The radial behavior of the hydrogenic eigenfunctions in position and momentum space is exponential and "Lorentzian", respectively, and their nodal structure depends on the associated Laguerre and Gegenbauer polynomials, respectively: RH(r)
--
2n(n + l)!
-~-l-~
(4)
215
N
R.(p)
=
(-i)~
p
2 Z ( n - l - 1)' l!(Z/n)t+2(4p) t+x p t+x 7r(n -4-l)! -((--'~n)2;p~)l+2 "'n-l-1
(p2/zjn 2) p -t- (Z/Tt) 2
(5)
The momentum-space expression was first given by Podolsky and Pauling [3] in 1929. Note, that for any real position function RH, the corresponding momentum radial function RH will be either purely real or purely imaginary, depending on whether the angular part of the orbital is even or odd (see also [8]). The factor ( - i ) l in Eqn.(5) has, e.g., the consequence that s-type and p-type functions do not "mix" in momentum space, which leads to hybrids that have a different nodal structure. 3. H Y B R I D S I N M O M E N T U M 3.1.
Hybrids
SPACE
of the spa-type
The most commonly encountered hybrids in organic chemistry are linear combinations of s and p-type orbitals. Depending on the linear coefficients of the real p-functions in the three Cartesian directions, the resulting set of hybrids will be oriented at different solid angles from each other. For example linear combinations of the form 1/2(s :t:px :t:py-t-pz) will yield hybrids with a tetrahedral angle among them. This basic geometry will be retained in momentum space, since the Fourier-transform (1) is a direction-preserving, unitary transformation. However, the fact that even and odd contributions to the positionspace hybrid are transformed separately (via cosine and sine transforms) into real and imaginary parts of the momentum-hybrid, means that the resulting densities (i.e., the square magnitudes) are inversion symmetric with respect to the origin, which is obviously not the case in position space. Therefore, an arrangement of orbitals in a point group G in position space, will lead to an arrangement in the point group G' = G x Ci in momentum space; this is the direct product of G with the inversion-symmetric group (see [7] and [9]). For the following considerations, the orientations of the hybrids are not relevant, and we therefore mix only pz orbitals with s-functions, resulting in an orbital that points along the polar axis z in both spaces. This takes the following form for the n = 2 shell:
_ ~
+
-
[2-
+
co
0]
=
,
(6)
S i~r(1 + a) r
= -~
(r
In position space, r 1 + v/-acos 0 r - - 2 Z(a cos 2 0 - 1)"
+ x/~r
- 16 Z s/2 [4p 2 - Z 2 - 4 ix/-dZpcosO] 7 r ~ + a ) ( Z 2 + 4p2) 3
(7)
is real and its nodal surface is defined by
(8)
In contrast, the real part of the momentum hybrid ~,~ has a spherical node (at p -
~ r ~ } _ Z/2), whereas the imaginary part has a planar one (at 0 - 7c/2). This means
216 ~a
that the subspace, where r - 0, is one-dimensional. It is, in fact, a circle of radius p = Z / 2 in the xy-plane 2, with center at p = 0. Coulson and Duncanson [4] obtained an analytical expression for a momentum hybrid of a C-atom, constructed from STO's that shows similar structure. Later, Cooper and Loades gave contour plots of several similar hybrids in momentum space [6]. However, neither one did comment on this interesting feature. It is widely known that total momentum densities for atoms are not always monotonically decreasing [10]. In fact the degree of non-monotonicity is dependent on the degree of p-population in an atom. This fact is visible as well in the shape of s p a hybrids in momentum space.
Figure 1. Surface plot of the orbital densities for sp a hybrids in momentum space. The hybrids are based on the hydrogenic wave functions. The three plots pertain to a = 1, 2 and 3, respectively. The hybrids point in the z-direction. A section through the density in the xz-plane is displayed.
In Fig.(1), we display the momentum density contributions of commonly encountered hybrid orbitals, obtained from hydrogenic eigenfunctions with Z = 1. The figure shows surface plots of the densities for s p a in the xz-plane for a = 1, 2 and 3. It may be seen that, while the s p hybrid exhibits a maximum at p = 0, greater p-contributions flatten this maximum out, leading to a plateau for s p 2, and finally a saddle point for s p 3. All of these densities feature two points in the xz-plane where the density vanishes exactly. They are situated on the x-axis, as sections along that axis demonstrate clearly. We show those in Fig.(2). Independently of the mixing coefficient a, those "nodal points" occur at x = -1-1/2 on each equatorial axis. They are the intersection of the aforementioned nodal circle with the displayed plane. To further assess the extremal structure of the densities corresponding to our hybrids, we also display the curvature of the density along the polar axis in Fig.(3). The analytical expression for these curves is 2Throughout the article, we use the notation x, y, z for the Cartesian components of the momentum vector p, and 0, r for its angular spherical coordinates. This is done to avoid excessive subscripts, and confusion with p-orbitals.
217
027r~p(ff) I
8192(a
-
2)
p=o = Z57c2( a + 1)
OP2z
(9)
Therefore, clearly the sp2-hybrid is the limiting case, for which the transition from maximum to saddle-point occurs, independently of the nuclear charge Z. We note that in the paper of Cooper and Loades [6], this distinction was not found for the momentum hybrids. Using Gaussian orbitals, with exponents tailored for the carbon atom, the origin is a saddle point for all three hybrids. For the hybrids that we have constructed, maxima will occur for any a > 2 at
Z Pmax -- ~-
(6 - 10a + 2x/i " 26a + 25a 2) 1/2
(10)
As we should expect, increasing the amount of p character, by increasing the value of a, will shift the maxima of the momentum distribution, to higher values of p. This fact well illustrates Epstein and Tanner's H y b r i d Orbital Principle, which states that "increased p character in an s - p type hybrid orbital results in increased density at high momentum". [11] In addition, one should contrast the appearance of the plots from Fig.(1) with our more familiar position space representation of these sp a hybrids, as shown in Fig.(4). There, we always see two distinct maxima [6], the sharper of which is located right at the origin.
3.2. Hybrids involving d-orbitals For most of organic chemistry the description in terms of sp a hybrids is sufficient for a qualitative picture. However, if the coordination numbers involved are greater than 4, as is the case for the majority of compounds involving transition metals, d-hybridization has to be taken into account. Since the m-quantum number of a d-function influences not
Figure 2. Sections through the orbital momentum densities displayed in Fig.(1) along the x-axis with z = 0, i.e. perpendicular to the main axis of the hybrid. The three plots pertain to a = 1, a = 2 and a = 3, respectively. Note that each density vanishes at two points on the x-axis.
o-1I
o-1I
0.08
01I
0.08
0.08
0"02 I -1
-0.5
oo '
-
-
.
oo
oxs
1
~
5
'oo
218
Figure 3. Second derivative of the orbital momentum density, 027r~p(ff)/Op~, in the z-direction, along the polar axis. Note that for a = 1 (left plot), this quantity is negative around p = 0, for a = 2 (middle plot) it is exactly zero, indicating a plateau, and for a = 3 (right plot) it is positive, denoting a saddle point.
ioo
-
2oo~
Figure 4. The sp, sp 2, and sp 3 hybrid density functions in the xz-plane of position space. As is often the case, orbitals that are quite different from one another in momentum space, can appear very similar in the corresponding position space representation.
only the orientation, but also the shape of the constituting atomic orbitals, we have to distinguish several cases. In the following, we focus on two of them; the first are hybrids that are directed along the z-axis, such as the ones in an octahedral complex, which have the general form 3sp~dbz2. The second lies in the xy-plane, and is exemplary of a hybrid that would be used to describe bonding in square-planar compounds. These hybrids are of the form 2 spxax2_y~ a-b . Other combinations are possible, but will in general show similar structural features.
219 The first class takes the following form (for the n = 3 shell) in momentum space:
~3sp~d2-a,b=
x/'l +la + b (~3s (#) + k/cd~3p~ (#) + k / ~ 3 d 2 (#))
18 3x/3--Z~ (81v/-2p4
--
6Z2p 2 [5~/-2 + 4x/b (3 cos20 - 1 ) ] + v/2Z 4)
(7rV/1 + a -t-b) (Z 2 + 9p2) 4 i 432pcosOv/Z7a
+
(Z 2 - 9p 2)
(~rv/1 -'~ a --}-b) (Z 2 nt- 9p2) 4 "
(11)
In the above equation, the hybrid is clearly broken down into a real part (second line), and an imaginary part (third line). We have found it convenient to analyze these two parts of the hybrid separately because of the earlier mentioned property that the real part of a hybrid will not "mix" with the imaginary part when computing expectation values and densities. The real part of Eqn. (11) arises from the mixing of s- and d-contributions. It has roots on either side of the xy-plane, on two closed C-rotation-symmetric surfaces, that are
Figure 5 9 Nodal surfaces of a s~3d ~'z 2z2 hybrid orbital with Z = 1 in the momentum-space representation. The left-hand plot contains two surfaces. One is the spherical node of the imaginary part. The second more complex surface consists of two closed and flattened spheres. These are the nodal surfaces belonging to the real part of the hybrid and are aligned along the z-axis. The intersection of the two types of nodes are two circles around the z-axis. The right-hand plot displays a cut through the xz-plane. Note that the (polar) z-axis is the horizontal axis in this plot. To avoid confusion, the nodal planes of the imaginary part are not displayed in either graph.
1IX f .. ...............~.5
"'\ z /,," ....................
-0.5
220 the solutions of a quadratic equation in p2. Figure (5) shows these surfaces (left plot), as well as a section through the zz-plane (right plot). The central sphere in the left plot and the circle in the right plot are nodes of the imaginary part of the hybrid. This imaginary part (third line of E q n . ( l l ) has nodal surfaces which consist of the xy plane (0 = 7r/2, not shown in the plots), and a sphere centered at p = 0 with radius Z/3. As ~a b a result, the density (i.e. the square magnitude of Cs;d,z), vanishes on a pair of circles at pz = =i=z [3 ( 1 - ~ r 1,2 with radius gz [ 2 / 3 + x/~/6v/~l U2 These circles are the intersection of the roots o t the real and the imaginary parts.-This nodal behavior is in clear contrast to the one in position space, where nodal surfaces of a rather complex shape are observed.
Figure 6. Surface plots of the momentum density corresponding to two different hybrids of the 3 b form sp~dz2. For the left plot, b = 2 (as in octahedral hybrids), for the right one, b = 5 (maximum bond strength). The surface plots are sections through the xz plane. Note that they both exhibit four points where the density vanishes exactly.
Fig.(6) shows the momentum densities corresponding to two different hybrids of the type. The first one is a sp3d 2 orbital as is encountered in octahedral complexes, the second one is a "maximum-bond hybrid" sp3d 5. The basic features are rather similar, although somewhat more strongly pronounced in the one with the greater d-component. The plots show a section through the densities in the zz-plane. Particularly for the sp3d 5 hybrid, a set of 4 "holes" parallel to the x-axis can be observed, arising from the aforementioned circular nodes. To show these "holes" more clearly, we have plotted in Fig.(7) circular sections through the densities, passing through the xz-plane with radius 1/3. The graphs show the momentum density as a function of the polar angle 0 in units of 7r. Note that the density is maximal inthe z-direction (0 = tTr;t = 0, 1,2), and in x-direction (0 = tTr;t = 1/2, 3/2), s -pz%~ a ..tb
221 Figure 7. Circular sections through the momentum densities displayed in Fig.(6). The plots display the value of the density along a circle of radius 1/3 in the xz-plane, as a function of the polar angle 0 -- tr. The nodal points are clearly visible.
, / 0
0.5
1 t
;0
I 5
k,o .
as would be expected and is also observed in position space. However, at angles of 00 and
[
7 r - 00 with 00 = a r c c o s ( ~ 2 ), the density goes exactly to zero, since this is the angle under which the real and imaginary nodes intersect. This angle is not dependent on Z, but depends weakly on b. It varies between 7r/2 for b = 1/2 and ~ / 2 - a r c s i n ( 1 / x / ~ ) for b ~ e~. The second class of spd-type hybrids that we will treat here are situated in the equatorial plane, i.e. at 0 = ~-/2. They consist of a linear combination of s, px and d~2_y~, orbitals, and have proven useful in describing the bonding in square-planar complexes. Their form (in momentum space) is:
Csp~d~2_y2-ab' (P) = X/'i +la + b (~3s(~[~) -3t-v/a~3p~ (]~) -+- v/b~3dz2-y2 (g)) 18Z 5/2 (x/~ (81p 4 + 2 4) - 622p 2 (12~/bcos(2r
5v/-6))
(~x/1 + a + b) (Z 2 + 9p2) 4 i 432 px/-Z-~ sin 0 cos r (Z 2 - 9p 2)
+
(~x/'l + a + b ) (Z 2 +
9p2) 4
(12)
Note that the imaginary part is essentially the same as in Eqn.(11), but the real part differs in shape. However, the consequence is a qualitatively different shape of both the nodal surfaces for the real part, and the nodal curves for the density. Figs.(8-10) show various aspects of the resulting density contribution for a 3sp~dx2_y2
222 Figure 8. Plots of the nodal surfaces in a sp~dx2_y2 hybrid orbital (Z = 1) in the momentum representation. The left plot shows the surface due to the real part (i.e., s and d contributions) only, whereas the right one combines it with the planar and spherical nodal surfaces characteristic of the imaginary (i.e., p-) component.
orbital, as this is the one used to describe square-planar complexes. Since the real part of Eqn.(12) is a linear combination of spherically symmetric s-contributions (with 2 spherical nodes) and "Rosetta-shaped" d-functions the resulting nodal surface is somewhat "donutshaped" around the y-direction. It is displayed in the left plot of Fig.(8). On the right-hand side we combined this surface with the planar and the spherical node of the imaginary part, which are due to the hybrids' p-contributions. Cuts through the xy-, xz- and yz-planes may serve to clarify the topology further (see Fig.(9)). The features in the two planes that are cut by the donut (xy and yz), are rather similar to the ones encountered earlier for the axial hybrids s P-a'b z t t z2. However, in the plane of the ring (xz) it differs considerably, and shows no intersections. All curves are rather contained within each other. Note also that the yz-plane is in itself a nodal plane of the imaginary part, and that therefore the cuts arising from the real part are nodes of the density. The intersections of the circle and the "outer" curves in the right plot are in fact intersections of 3 nodal surfaces, 2 from the imaginary and one from the real part. The lower right plot of Fig.(9) shows the nodes of the density (i.e., the intersections in 3Dspace). The two closed curves in the y z-plane arise from intersections of the donut-shaped node of the real part with the planar node of the imaginary one, whereas the other curves are intersections of the spherical node of the imaginary part with the donut, close to its "hole". The explicit expressions for the curves displayed in the lower right plot can be derived. They are: Z
p--~(V/4-x/~sin20•
;r
•
(13)
223 Figure 9. Traces of the nodal surfaces in the xy-, xz- and yz- planes for the 8p2dx2_y2 hybrid. It can be seen that while the basic features in two of the planes (xy and yz) are rather similar to the one observed in 8-ad pz b.2 type hybrids, the situation in the xz-plane is completely different. The lower right plot shows the nodal lines that are the intersections of various surfaces displayed in Fig.(8).
9f /'/l(,~ .... ',, -~ t '~ ;~176176
\.._ .....~
x
........./
-0"5t -1
1Y z
0.5
/
(o2
z \ . . . . . .
4
x
O.
-0.
t,.,
-0"5 I -1
r
2
2arcc~
V~sin2 0
;p=
It is rather surprising that such simple linear combinations will produce such complicated topologies in momentum space. Fig.(10) shows a surface plot of a section of the momentum density in the xy-plane, where the density is accumulated (left). The seemingly monotonous distribution exhibits on closer inspection a good deal of fine structure: first, there are the aforementioned "holes" in the vicinity of the nodal lines; secondly, the apparent maximum reveals itself on an enhanced scale (right plot), to be a saddle point that is minimal in the x-direction.
224 2 Figure 10 9 Surface plot (left) of the momentum density corresponding to a spxdz2_y2 hybrid orbital. The section displayed lies in the my-plane. Although the overall features seem to be rather weak, a complicated nodal structure is observed, and the origin is a saddle point (right). The right plot shows the momentum density on the x-axis.
49.25
-0.03
-0.02
-0. Ol
0
0. Ol X 0 . 0 2
0.03
For all momentum densities, the origin of momentum space is necessarily a critical point. This arises from the inversion center and the requirement of continuity. However, the topology at that point may vary considerably for the hybrids considered here. We pointed out already in the previous section that the spa-hybrids do not always exhibit a pair of off-center maxima. For hybrids containing d-functions, the picture is further complicated 9 To obtain a clear idea, it is best to obtain the diagonal elements of the Hessian matrix of the density Hij = 02rr~(p-)/OpiOpj in Cartesian coordinates 9 From their sign it may be inferred what type of critical point is observed. For the sp~dbz2-type hybrids, this yields
(Oq27r~bOq27rrOq271-~b)(2~/~ 11 2V/~ 0p
'op--7'0p
-
V27rr
=
-
r ( 8 a - 33)
'
-
l18a '
4V/~ -
11)
(15)
-
(16)
where r = 46656/Z57r2(1 + a + b) is a common pre-factor. The eigenvalue structure in Eqn.(15) indicates that p = 0 is a minimum in z-direction, whenever b < 2a 2
lla 121 --~+--~
and
a > 11/8,
(17)
These conditions are independent of Z, since the latter determines only the spatial extent, but not the basic topology. From Eqn.(15) we also infer that d-contributions tend to flatten out the minimum at the origin (see also Fig.(6)) by moving the maxima towards higher momenta, whereas
225 Figure 11. Map for the topology of the momentum density at the origin of momentum space p 0, for hybrids containing d-functions. The left plot is for functions of the Spzadbz2-type, the right one for the Spzadb~2_y2-type. The axes are the mixing coefficients a and b, respectively, and the lines drawn separate regions of different topology. =
35
60I
30'
5O
:o
ring
~~t
/
i0_
25' saddle
20 b 15
rlng
minimum
.
I0
l
5 maximum ,
O'
.0,0
~~ '
'
i
,
J ---~, .... 2
a
saddle ~
3
,
~1 . . . .
p-contributions sharpen the minimum, although not deepening it. In the x-direction (or 121 more generally, perpendicular to the polar axis), p = 0 is a maximum for b < - U , i.e. in all cases of practical interest. The left plot in Fig.(ll) shows the regions of different possible topologies at p - 0 in the (a, b) plane. Note that for "reasonable" hybrids, a < 3, and the origin in momentum space is therefore either a saddle point or a maximum. Rings or minima would only occur with very low relative s-contributions. Note that the Laplacian (16) (as the sum of second derivatives) at the origin does not depend on the d-contributions. They merely serve to "rearrange" the density, whereas p-components will make the Laplacian less and less negative. For the other type of d-containing hybrids, the spxdx2_v2 hybrids, we have
op~'
Op~'
Op~ v~r
~-
'
~ (Sa - 33)
11' -11/ (19)
where T is the same as in Eqn.(15). This means, the density is always a maximum in the z-direction at p = 0 for this type of hybrid. In the x-direction (which is the main axis of the hybrid), it is minimal if b<~8a2
22a3 ~ 12124 and
a > 11/8,
(20)
and in the y-direction it is maximal as long as b < 121/24 ~ 5.0417, i.e. up to and including spxads~2_y2 hybrids The Laplacian is the same as for the previous type, which is
226 a trivial consequence of the fact that it depends (for p = 0) only on the radial part of the constituting orbitals. Note that our example (sp~dx2_y2) has a saddle point of the momentum density at the origin. This fact can not be seen at the resolution of Fig.(10), since the saddle is extremely shallow (the second derivative is "only" 11664(5- 2v/g)/Tr 2 ~ 911.4. The right-hand plot in Fig.(11) shows the relevant section in the (a,b) plane with the proper designations for the type of critical point occurring at the origin. Note that a cage-structure is impossible since the hybrid is always concentrated in the xy-plane. Table 1 Moments of the sp a hybrids in both coordinate and momentum spaces. (r ~) n--
2
1 (3+a)Z 2 12 l+a
n--
1
1Z
128 a+2 15 ZTr(l+a)
6+5a
16 Z(3+4a) 45 7r(l+a) ,,,
4 39+7a
3 Z2(l+a)
n--O n--1
z(i+~)
n--2
6z2(t+~)
n--3
30 Z3(l+a)
n--4
240 Z4(l+a)
5a+7
1 2 ~Z
11+7a
12+7a
15 7r(l+a)
1_z4(37+7~) 48
l+a
4. M o m e n t s of the H y b r i d Orbitals
It is a simple matter to derive expressions for the moments of the hybrid orbital densities. In momentum space, the expressions will take the form, f pnrr~(g)dg = f ({a})Z n
(21)
where f ({a}) is a function of the mixing coefficients ( one mixing coefficient for the 8p a hybrids, two mixing coefficients for the sp~d b hybrids, etc.). The analogous expression in position space has an inverse dependence on the atomic charge, Z.
/ r'~pr
" - g ({a}) Z-~
(22)
where g ({a}) is also a function of the mixing coefficients, although different from the f function in Eqn. (21).
227
Complete expressions for the hybrid charge density moments, in both position and momentum space, are given in Tables 1 and 2. The former table contains results for the sp a hybrids while the latter gives results for the spad b hybrids. Table 2 Moments of the spa d b hybrids in both coordinate and momentum spaces.
(p')
(<) 2
n -- --2
n-
405
(15+3b+5a)Z 2 1+aTb
9 105+9b+25a
5 Z2(l+a+b) 16 200a+128b+335 175 ZTr(l+a+b)
1
n-0 8 Z(135+160aq-192b)
1 27+45a+21b
n-1
i575
2 Z(l+a+b) 20a+14b+23
Ir(1-t-a+b)
9 Z2(l+a+b)
1Z 2
n-3
81 70a+42b+85
16 Z3(200a+128b+335) 14175 rr(l+aTb)
n-4
405 238a+126b+303
n-
2
2 Z3(l+a+b) 2
Z4(l+a+b)
1 Z4(lO5-t-9b+25a) 405
l+a+b
Notice that the (r -1) and the (p2) moments have no dependence on the mixing parameters. They are simple functions of Z. This is a feature unique to the case where the hybrid orbitals are constructed from degenerate atomic orbitals. Since hydrogenic functions with the same n quantum number fulfill this condition, the "energy" moments, (r -1) and (p2}, will not depend on how the hybrid orbitals are mixed. The moments in Table 2 also do not depend on what types of d functions are used to construct the hybrids. In other words, the moments do not depend on the m quantum number. For example, a hybrid which uses only dz~ orbitals, will have the same moments as a hybrid which uses only dx2_y2 orbitals. This is, perhaps, an obvious point as the moments we are discussing are radial moments; angular characteristics in the hybrid density are spherically averaged during the integration over the solid angle. To see this more clearly, note the form of the integral (pn) for the case of an sp a hybrid.
(pn) --1
1 +a S (R2,o(P)Yo,o+ V/-aR2,I(P)YI,m)9pn (/~2,0(P)yo, 0 -t- %lraI~2,l(P)Yl,m)dp
(23)
The orthogonality of the spherical harmonics, Y/,m, insures that there will be no mixing of radial terms, Rn,t, in the resulting integral. Since the spherical harmonics are also normalized to unity, simplification of Eqn.(23) will produce the following functional form.
(P'~)spo h y b r i d
--"
1 +1 a
/
+ aR ,l (p)*p'R%I(;)] dp
(24)
228
(25)
l+a
Therefore, the hybrid density moment is simply a properly weighted average of each individual orbital moment. The orthogonality of the spherical harmonics has insured that different radial functions do not mix together. An analogous expression holds for the radial moments of the sp~d b hybrids.
1
(pn ) spadb hybrid -- 1 ~- a Jr b ( (pn ) s ~
-~- a (pn ) p ~
-~- b(pn ) d ~
)
(26)
5. C o n c l u s i o n In 1929, Linus Pauling, together with Boris Podolosky, became the first person to publish the m o m e n t u m representation of the eigenfunctions of a single-particle Coulombic Hamiltonian. Although he did not publish any more work on momentum space concepts, he is nonetheless a pioneer in the field of Momentum Space Quantum Chemistry. In this work, we brought together two of Pauling's contributions to Chemistry. The first, as mentioned above, is his pioneering work in momentum space quantum mechanics. The second is the concept of hybrid orbitals, originally used to understand the strengths and directional characteristics of covalent bonds. Accordingly, we have combined these two areas by looking at the form of some of the most useful hybrid orbitals, in momentum space. Among the more interesting qualities of momentum space hybrids, is the lack of strong directional asymmetry, this being one of the most noticeable characteristics of position space hybrids. Indeed, it is this directional asymmetry which has made hybrid orbitals so useful for describing directional bonds. In momentum space, however, the hybrids are inversion-symmetric and this is shown to have a profound effect on the nature of these orbitals. Another difference is the nodal structure of these atomic contributions to the total density. The hybrid orbitals as we know them, in position space, exhibit nodal surfaces, i.e. two-dimensional subspaces on which the density vanishes. This dimensionality is reduced in momentum space. Here, the nodes are invariably one dimensional, i.e. curves that are formed by the intersection of real and imaginary nodal planes. Finally, (for atoms), the momentum densities corresponding to hybrid orbitals exhibit a few basic extremal features close to the origin. These depend on the weight that is given to s, p and d contributions, and they determine the basic "look" of the density. Outwardly, momentum-space hybrids share one feature with a related experimental quantity, the Compton profile: they all look alike. On closer inspection, however, there are a variety of complex features, mainly arising from the nodal structure of the orbitals. Apart from the obvious use of hybrids in position space for the description of bond situations, there is another feature that has always captured the interest of scientists and laymen: their intricate structure. This feature is less apparent in momentum space, but it is still present. If nothing else, its enjoyment makes a close look at these entities worthwhile.
229 6. Acknowledgments We are grateful to Dr. Jiahu Wang and Minhhuy H6 for fruitful discussions, and to Dr. Zelek Herman for making the Pauling bibliography available to us. REFERENCES 1. Z. S. Herman. Some early (and lasting) contributions of Linus Pauling to quantum mechanics and statistical mechanics. In Molecules in Natural Science and Medicine: An Encomium for Linus Pauling, Z. Maksid and M. Eckert-Maksid, Eds. Ellis Horwood Limited, West Sussex, England, 1991, pp. 179-200. 2. Linus Pauling. The Nature of the Chemical Bond. Cornell University Press, Ithaca, New York, 1948. 3. B. Podolosky and L. Pauling, Phys. Rev. 36 (1929) 109. 4. C.A. Coulson and W.E. Duncanson, Proc. Cambr. Phil. Soc. 37 (1941) 67. 5. A. Rozendaal and E.J. Baerends, Chem. Phys. 95 (1985) 57. 6. D.L. Cooper and S. D. Loades, J. Mol. Struct. 229 (1991) 189. 7. P. Kaijser and V.H. Smith Jr. In Methods and structure in quantum science, J. Calais, O. Goscinski, J. Linderberg, and Y. Ohrn, Eds. Plenum Press, New York, 1976, pp. 417-426. 8. P. Kaijser and V.H. Smith Jr. Evaluation of momentum distributions and Compton profiles for atomic and molecular systems. In Advances in Quantum Chemistry, vol. 10. Academic Press, New York, 1977, pp. 37-76. 9. S.R. Gadre, A.C. Limaye and S.A. Kulkarni, J. Chem. Phys. 94 (1991) 8040. 10. W.M. Westgate, A.M. Simas and V.H. Smith Jr., J. Chem. Phys. 83 (1985) 4054. 11. I.R. Epstein and A.C. Tanner. In Cornpton Scattering, B. Williams, Ed. McGraw-Hill, New York, 1977, pp. 209-233.