Use of Electron-Density Plots in Applied Quantum

Use of Electron-Density Plots in Applied Quantum Chemistry By Ilyas Absar and John R. Van Wazerr] The uses in applied quantum chemistry of computer-produced three-dimensional diagrams of electron-density distributions are discussed. The two major catagories of such diagrams are contrasted; and advantages are described for plots showing as the third dimension the point-by-point electron density in a cross-sectional cut (corresponding to the other two dimensions) through a molecule. Examples are presented to show how these plots may be used (1) to assess the adequacies of a given mathematical representation employed in the calculation of a wavefunction, (2) to clarify the interrelationship between an individual molecular orbital and the atomic orbitals of the constituent atoms, as well as (3) to explain the canonical set of molecular orbitals of any selected molecule. Furthermore these plots serve (4) to demonstrate clearly a replication of key characteristics between certain molecular orbitals in different molecules. These examples are accompanied by others which indicate how the electron-density plots may be used (5) to understand and clarify accepted chemical dogma. Finally, the possibility is discussed of employing quantum calculations on a wider scale so as to be of value in the more practical aspects of chemistry.

1. Introduction Although electron-density diagrams have been used for many years in quantum chemistry, several kinds of threedimensional plots accurately produced from wavefunctions by computer-controlled graphic systems have been introducedr'-41 since 1970. It has turned out that such plots are useful aids in interpretingr5I the results obtained from quantum calculations. Several booksC5- 71 treating this matter present many examples of such plots for various atoms and molecules; and it is our purpose here to survey these plots and to review some of their uses other than the obvious one of just depicting data.

2. Choice of Plots

+,

Since a wavefunction, or its corresponding electron density, must be described in four dimensions (three for the spatial geometry and one for the point-by-point value of the function), any three-dimensional plot must leave out some information; but yet no more than three dimensions may be depicted clearly in a spatial model or in its corresponding perspective diagram plotted on a sheet of paper. Two ways of handling this problem are commonly employed. One is to show the variation of the function across a chosen plane (or other surface) passing through the subject array of atoms, e.g. a molecule or crystal. In a linear or planar molecule, the chosen plane is generally selected to be that of the atoms; but, even for planar molecules, it is also often informative to plot one or more additional planes perpendicular to the plane of the molecule. For more complex molecular geometries, judicious selection of the planes to be studied must be made in order to depict clearly the spatial variation of

++*,

[*] Dr. I. Absar Department of Mathematics, Queens University Kingston K7L 3N6 (Canada) Prof. Dr. J. R. Van Wazer Vanderbilt University Nashville, Tennessee 37235 (USA)

80

the wavefunction or electron density. These cross-sectional plots have been preparedr6] using a linear or logarithmic scale for the electron density. Although the logarithmic scale (or a square-root or cube-root scale) has the advantage of encompassing the highest density values while simultaneously emphasizing the lower values appearing in the bonding and intermolecular regions, it does give a distorted density presentation. We prefer''] to use a linear scale for the electron density and to truncate the sharp density peaks that appear close to the nuclei of atoms having core electrons. The other way is to show only the surface(s) corresponding to some chosen value of the function in space so as to obtain what we usually call shape plots. The difficulties here are that the detailed spatial distribution of the function is thereby virtually ignored, while the selection of the chosen value of the function is extremely arbitrary. This latter problem is illustrated in Figure 1, in which a cross-sectional electrondensity distribution (using a linear scale) for water sits on top of a series of shape plots corresponding to various chosen values of the electron density in this molecule. It is clear from these shape plots (a through e) that the choice of a relatively high electron density for this molecule will give only a tiny, nearly perfect sphere closely surrounding the oxygen nucleus; whereas a very low value will lead to a large, nearly spherical contour centered on the molecule. Thus, details of molecular structure will show up in shape plots only within a restricted intermediate range of the adjustable parameter (the chosen electron density). O n those few occasions in which we find it valuable to calculate shape plots, we commonly employ a value for this parameter of O.6e/A3 for depicting the whole atom or molecule and 0.14e/A3 for the constituent valence orbitals, since these choices were foundr5I to correspond rather well to the van der Waals radii of common atoms from the second and third period of the Periodic Table. This choice of 0.14e/A3 was employed in the shape plots describing the molecular orbitals of waterL5] in Figure 2. Because of the detailed point-by-point information presented in cross-sectional electron-density distributions, this Angew. Chem. Int. Ed. Engl. 17,80-88 ( I 9 7 8 )

0

n

Fig. 1. Comparison of a cross-sectional electron-density plot (above) for the atomic plane of the water molecule with a series of shape plots. Shape plots a-e correspond to cuts through the cross-sectional diagram at the heights indicated.

kind of linearly scaled plot will be shown in most of the examples presented below to illustrate advantageous uses of the computer-produced three-dimensional graphs. Although it is possible to produce similar cross-sectional plots of wavefunctions $, we have decided not to include this approach among the following examples, since electron density $$* is a physically meaningful quantity while the wavefunction is not.

lity giving lower (i. e. more negative) energies. This approach however may sometimes be misleading, so that a low-energy result can be of much poorer quality than one of appreciably higher energy. For example, we have intercompared['] crosssectional electron-density plots of the canonical molecular orbitalsY1 of the water molecule calculated in a variety of basis sets. These plots showed, as expected,that an extended Slaterl8] and a large Gaussian basis set[g](both with polarizing functions) gave electronic distributions that were essentially identical and that the quality of much smaller Slater and Gaussian descriptions was quite good, except for a readily noted rounding of the top of the sharp humps corresponding to the hydrogen atoms in the Gaussian basis set. Indeed, even a CNDO semiempirical calculation[' '* I '1 gave quite well-shaped electronic distributions for each of the valence-shell molecular orbitals. However, as exemplified for orbital 1 b2 in Figure 3, a one-center calculationI'21 employing an extended-Slater basis set centered solely on the oxygen atom did very poorly in that there was no evidence whatsoever of the expected humps at the site of each hydrogen atom. This behavior was observed for all of the valence orbitals (2al, 1b2, and 3al) that ought to show clumping of electronic charge around each hydrogen. It was also seen for the total electronic density. The total energy was calculated to be -75.922 a.u. for this one-center representation as compared to -76.005 a.u. for the extended-Slater and only -75.703 a.u. for the minimum-Slater basis setr"] (which gave properly shaped electronic distributions). About 15 years ago, one-center calculations were very popular because (1) they were less costly than the usual calculation in which appropriate atomic orbitals were assigned to all of the atoms in the molecule and (2) the resulting total energies for a given level of adequacy of the basis set were very good for the molecules being studied at that time. On the basis of this one example, it would seem that the reason the one-center representation worked so well was that binaryhydrides (such as methane, ammonia, and water) were emphasized in theoretical studies at that time. Because of the small energy contribution to a molecule from a hydrogen atom as compared to a second-period atom, a good basis set so improved the contribution of the latter to the total energy that neglect of the hydrogens was more than offset. Incidentally, most of the total energy of water is attributable to the presence of the oxygen core electrons. We suggest that anyone investigating an alternative method for calculating wavefunctions would be well advised to employ plots of the type described herein to see if the spatial distribution of the electron density (or the wavefunction) exhibits at least the proper gross shape. During the last few years, one of us ( J . !T K) and his associates have been devoting considerable e f f ~ r t [ ' ~ - 'to ~] making the pseudopotential technique for solving the Schroedinger equation into a practical method (NOCOR) for use in everyday quantum chemistry. Therefore, quite early in this endeavor we compared cross-sectional electron-density plots of the valence shell for the NOCOR wavefunctions against those from an equivalent full-SCF calculation, as shown in

3. Adequacy of Various Mathematical Shortcuts

p] The canonical molecular orbitals are understood to be that

The usual off-hand criterion for the adequacy of a basis set employed in a self-consistent-field (SCF) calculation is the calculated total energy, with wavefunctions of higher quaAngew. Chem. Int. Ed. Engl. 17.80-88 ( I 978)

set among the infinite range of possible MOs which represents the solution of the Hartree-Fock SCF matrix and is adapted to the symmetry of the molecule. Cf. W Kutzetnigg, Angew. Chem. 85, 551 (1973); Angew. Chem. Int. Ed. Engl. 12, 546 (1973), especially p. 559.

81

I

\

0

'b2

Fig. 2. Shape (left) and cross-sectional electron-density diagrams (right) for the valence-shell molecular orbitals of water.

Figure 4. In the usual NOCOR procedure, the valence electrons are handled by the standard SCF procedure while the set of core electrons of each atom are simulated by a potential function and the core-valence interactions are treated by the pseudopotential function Vp [eq. 11, in which E , and 4, represent the core eigenvalues and eigenfunctions, while the subscript v refers to a valence orbital and xto is a pseudowavefunction representing this valence orbital.

(1)

In view of the fact that in this approach there are no orthogonality restraints between the individual core electrons and a valence electron, the wavefunction of the latter has no need to exhibit a nodal structure in the core region. Therefore the electron densities corresponding to the valence molecular orbitals shown for the I2 in Figure 4 smoothly die out in the core region in such a way that the integrated electronic density of each orbital in this region equals that 82

corresponding to the summation of the antinodes occurring in the same region in the case of the same orbital obtained from a full-SCF calculation. The point of Figure 4 (and of similar intercomparisons of electrondensity distribution plots from pseudopotential and full-SCF calculations) is that in the valence region the wavefunction of the total molecule or of each individual orbital is essentially indistinguishable from one obtained from an equivalent SCF calculation that includes all of the core electrons.

4. Interrelationships between Orbitals Since atoms have only one nucleus, the symmetry of individual orbitals (e.g. s, p, d, etc.) must be discussed in terms of these orbitals and cannot be treated with respect to the nuclear geometry. On the other hand, it is standard practice in the case of molecular orbitals to employ appropriate grouptheory designations derived from thc symmetry of the nuclear acrangement (e.g. al, az,and e for a molecule of Csvsymmetry). As a result, this scheme of notation for describing molecular Angew. Chem. Inr. Ed. Engl. 17.8&88 (1978)

Fig. 3. Cross-sectional electron-density plots of the l b 2 orbital of water: a) with an extended-Slater basis set: b) in a one-center representation; c) in a minimum Slater presentation; d) with a moderately-sized Gaussian basis set employing seven s- and three p-type exponents for the oxygen and three s-type for each hydrogen.

Fig. 4. Comparison of the electron-density plots of three valence orbitals (starting at the bottom, 19 og,21 crg, and 11 n.) of the I2 molecule as determined in a full-SCF calculation (left column) and by the NOCOR pseudopotential method (right column) in the same basis set. Note that the antinodes of the 5 s and 5 p contributing atomic orbitals appearing in the core region do not show up in the NOCOR calculation which, howeve&, beautifully reproduces the valence-bonding structure. Angew. Chem. I n t . Ed. Engl. 17,80-88 ( 1 9 7 8 )

orbitals does not make clear their constitution in terms of the atomic orbitals of the contributing atoms. However, the relationship between a molecular orbital and its contributing atomic orbitals may be seen at a glance from an electrondensity distribution plot. The connection is clearly seen in either of the two kinds of plots shown in Figure 2, where the most stable of the H 2 0 valence orbitals, 2al, is seen to be based on the 2s atomic orbital of oxygen. Each of the three remaining filled valence molecular orbitals of water is based on one of a set of three orthogonal oxygen p orbitals. The most stable of these three p-dominated orbitals is 1b2 in which the CZvaxis of the molecule passes through the nodal plane. This is followed by orbital 3al in which the nodal plane of the oxygen p orbital lies parallel to the line connecting the pair of hydrogen atoms. In the least stable of the filled orbitals, Ibl, the nodal plane passes through the two hydrogen atoms so that there can be no H-0 bonding. As a result, when going from the most to the least stable of the valence orbitals, one proceeds from a nodeless molecular orbital through a set of three orbitals exhibiting orthogonal nodal planes-a molecular sequence that is reminiscent of the stability sequence found for a second-period isolated atom, e . g . 2s and the three 2p's. Moreover, atomic orbitals are utilized in building molecular orbitals in such a way that the more stable atomic orbitals appear in the more stable molecular orbitals as long as the nodal requirements of the orbitals are met. In other words, one can consider molecular orbitals in terms of a modified aufbau principle, that is analogous to the one concerned with atomic orbitals. This aufbau approach holds for the molecular orbitals of highly complicated molecules of varying symmetries, and this is readily seen['] from the electron-density plots. In working with electron-density plots of molecular orbitals, we that each molecular orbital of a given molecule bore striking resemblances (with respect to the magnitude and geometry of the individual atomic contributions) to molecular orbitals of other molecules. Even though this close correspondence between certain molecular orbitals in different molecules was sometimes seen in quite unrelated chemical structures, the effect was first noticed and is particularly pronounced in the case of a series of related molecules. This has been demonstrated in our orbital for a number of molecular sequences such as methanethiol, methanol, and dimethyl ether; methylamine, methylphosphane, and methylphosphorane; cyclopropane, phosphirane, and thiirane; or water, formaldehyde, and ketene. Most of the time, orbitals shown to be interrelated by inspection of the electron-density plots follow the standard orbital-correlation rules of quantum chemistry. However there are occasional exceptions, each of which may be explained logically. As an example of these intermolecular relationships, consider the valence-shell molecular orbitals of H2S, H2S0, and HzS02, the latter two molecules being analogs of sulfoxides and sulfones. The underlying computations on the progressive introduction of one and then two S-0 bonds to H2S have been reported['*! Figure 5 shows the intermolecular relationships (dotted lines) between the molecular orbitals of HZS, HISO, and HzSO2.Although most of these relationships can be established unambiguously from symmetry considerations, orbital-density plots not only give a clear-cut demonstration of which orbitals across a series of molecules are interrelated 83

but also delineate the details of how this occurs, thus leading to the resolution of any ambiguities.

however the character of the charge distribution about the sulfur nucleus again persists, the similarity being strikingly apparent in comparing the two more symmetric molecules HIS and H 2 S 0 2 .

5. Electron-Density Difference Plots

il 2 15 0

-b2

._.. -2."

5

-88' 0

5

Fig. 5. A correlation plot for the valence orbitals of HzS, HZSO,and HzSOZ, showing the most stable orbitals at the top. The orbitals involved in S-0 and S-H bonding are so indicated; while 1.p. stands for the lone-pair orbitals, with their assignment to either sulfur or oxygen being denoted above the line corresponding to each orbital.

Two orbital sequences involving shifts from sulfur lone-pair to oxygen lone-pair character are depicted in Figure 6. The sequence of Figure 6 a (5al of HzS, 8a' of HzSO, and 7al of H 2 S 0 2 ) shows a persistence of charge attributable to the sulfur lone pairs in HzS, not only in HzSO but also in HzSOZ! It should be observed that the oxygen lone-pair charge in HzSO reappears practically unchanged for each oxygen of H 2 S 0 2 ; also the small charge concentration lying on the axis between the two hydrogen atoms persists throughout the entire sequence. In Figure 6 b are shown orbitals 2bl of H I S , 9a' of H I S O , and 4bl of H2S02. The transition from sulfur to oxygen lone-pair character occurs through conversion of sulfur lone-pair charge into S-0 bonding;

Variations in electron densities can be studied by the use of cross-sectional electron-density-differenceplots, which are perhaps an even more powerful tool than the density plots themselves. Although the commonly used difference plots are those showing the electron density of the molecule minus the overlapping electron densities of the unbonded atoms positioned as in the molecule, difference plots may be used in many other ways, some quite novel. An example of the common type of difference plot is shown according to our approach in Figure 7 for the ethane molecule, with the plane of observation passing through the two carbon atoms and a pair of (anti) hydrogen atoms['l. Figure 7a depicts the electron density of the CzH6 molecule minus that due to superpositions of the two carbon atoms in their usual szp2 ground state and the six hydrogen atoms, with these atoms centeredlat the proper positions corresponding to the ethane molecule and being described in the same basis set as the molecule. Figure 7b is a similar plot, except that in 'this case the carbon atoms have been hydridized to the sp3 configuration. Both of these difference plots show an accumulation in the C-H and C--C bonding regions due to molecule formation. In Figure 7 a the negative cones at the positions of the atomic nuclei clearly demonstrate that much of the change in charge between the atoms within and without the molecule involves the s orbitals of the carbon atoms. The major difference between the two plots is the sharp downward pointing peaks (Fig. 7a) and upward pointing peaks (Fig. 7b) at the carbon nuclei. This is due to the fact that, in the sp3 hybridization of the carbon atoms, one of the 2s electrons of the s2pz ground state was transferred to a 2p

Fig. 6. Electron-density diagrams for two selected sets of correlated orbitals in the series of molecules: HIS, HISO, and HzS02. a) orbitals 5al, 8a', 7a,; b) 2bl, 9a', 4bl.

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Angew. Chem. I n t . Ed. Engl. 17,80-88 (1978)

orbital. The plots show that in the ground state there is too much whereas in the sp3 hybrid there is not enough charge of the 2s type. Obviously, proper hybridization must lie somewhere in between. H

Fig. 7. Electron-density-difference plots for a plane passing through the two carbon atoms and a pair of (anti) hydrogen atoms in the ethane molecule. a) molecular electron density minus that obtained by summing over the ground-state constituent atoms placed in the same positions in which they appear in the molecule. b) as in a) except that the carbon atoms of the summation have been hybridized to the sp3 configuration. These diagrams are plotted to show both plus and minus differences, so that they appear to be transparent. Note that the electron-density differences in these plots are magnified to show up details.

lS

h

2s

Fig. 8. Cross-sectional electron-density plots of the sulfur core orbitals of s symmetry in the series of molecules H2S, HISO, and HzS02. Note that the 2s orbital exhibits appreciable deformation from its spherical shape in the isolated sulfur atom; whereas this is not true for the Is orbital. The positions of the hydrogen and oxygen atoms are indicated on one of these plots. Angew. Chem. I n t . Ed. Engl. I7,8&88 ( 1 9 7 8 )

An example of the novel usage of electron-density-difference maps is to elucidate very small changes in electronic structure, as was found in a study[’’j of the relationship between the inner-shell molecular orbitals of a series of structurally related molecules. Figure 8 shows electron-density-difference maps for the sulfur “1s” and “2s” molecular orbitals minus the respective atomic orbitals for H2S, HzSO, and H2S02. Note that the charge distribution of the sulfur “2s” core electron in any of these molecules no longer exhibits spherical symmetry. This effect is most pronoQnced in the “2s” orbitals of HIS and H2S0, the two cases where sulfur lone pairs are present. Figure 8 demonstrates that the core electrons are not as insensitive to changes in the valence structure as was supposed for so many years. Inner-orbital photoelectron spectroscopy (ESCA) has experimentally demonstrated this fact with respect to the core binding energies[20-22! The matter of small geometrical deformations in core orbitals (as indicated in Fig. 8) is worthy of further consideration in more precise calculations than the moderately sized basis sets we employed. Other interesting phenomena (such as the electronic effects of internal rotation about a particular bond, ring-closure reactions, and the significance of discussing particular bonds with respect to their covalent or ionic character) are fruitfully stuusing electron-density-differenceplots. A set of difference plots showing the effect of allowing d character to third-period atoms in a molecule is presented in Figure 10 (see below).

6. Clarification of Chemical Dogma There is a common misconception that SCF wavefunctions are too delocalized for direct interpretation. As a result, considerable effort has been put into for the localization of the canonical molecular orbitals resulting from singledeterminant SCF calculations. However, one of the important contributions to the understanding of molecular orbitals made by electron-density plots is a clear demonstration of the fact that canonical molecular orbitals are readily understood in terms of the constituent atomic orbitals and that their interpretation in terms of standard chemical theory is easy and straightforward. This has been demonstrated by our bookr5] and numerous papers from our . Although the numerical array of a wavefunction (or many of the various other mathematical arrays which can be used to represent it) may be utilized to characterize a molecular orbital with respect to the details of its atomic makeup, this approach is somewhat cumbersome and often not fully informative, even for those experienced in the mathematics of quantum chemistry. On the other hand, a quick visual inspection of a properly chosen three-dimensional cross-sectional electrondensity plot shows up the details with great clarity. The fact that a single-determinant wavefunction can be decomposed into an infinite number of different sets of “orbitals” has led some people to feel that molecular orbitals are highly arbitrary constructs. This is untrue however, since the molecular orbitals obtained from a standard SCF calculation (the canonical set) are just as meaningful as is the use of s, p, d, etc. orbitals for describing isolated atoms, and in the same sense. This means that anyone who uses the standard s-p-d nomenclature for describing the symmetry of atomic 85

orbitals must, for exactly the same reasons that these atomic orbitals were chosen over other alternatives (hybrids), put equal faith in the canonical molecular orbitals. Inspection of cross-sectional electron-density plots (e.g . see Figs. 2-9) shows how the canonical molecular orbitals retain much of the form of the constituent atomic orbitals. The o,x, and 6 notation used in discussing chemical bonding properly applies only to linear molecules. However, for many years chemists have applied this notation to molecules having other than linear symmetry, sometimes quite inappropriately. Another aspect of three-dimension cross-sectional electrondensity plots is that they show graphically how appropriate is the use of this notation for discussing chemical bonding in a given molecule and demonstrate the complications involved. For example, examination of a series of electrondensity diagrams corresponding to varying amounts of internal rotation of a molecule which does not exhibit a high degree of symmetry with respect to the bond axis of rotation has shown that in such cases one can indeed speak of twisting of the electronic distributions making up the bond. This behavior is demonstrated by the valence orbitals of diphosphane. As shown in Figure 9, an HzPPHz orbital[3n1which is x-like with respect to the P-P bond in the eclipsed and staggered forms of diphosphane, may be considered as being a twisted x bond in the intermediate rotameric forms in which the pair of hydrogen atoms on each phosphorus are no longer present in relative positions which would allow “normal” n-bonding[’l. In discussing the contribution of d orbitals to the chemistry of atoms of the Third Period, chemists have customarily differentiated sharply between p n - + dcharge n transfer and the polarization of d orbitals. When electron-density plots are made for most molecules (even those in which pll+d, charge transfer is to be expected), it turns out that electron-density plots of the entire molecule, the valence shell, or, to a lesser extent, the various individual orbitals as calculated with d character being allowed or disallowed to the charge-accepting atom differ only slightly. However, by subtracting the point-by-point electron density in, say, the no-d plot from that in the plot where d character was incorporated, one obtains an electrondensity-differenceplot in which the only features are the small changes from the one to the other of the electron-density distributions being compared. For example,in the case of the chlorosilane molecule, Howell and Van Wuzer carried out four SCF calculations[311:One corresponding to d orbitals being disallowed, another with appropriate d functions being incorporated into the mathema-

Fig. 9. Cross-sectional electron-density plots for one series of related orbitals obtained by internal rotation of the diphosphane molecule (H2PPH2).a) eclipsed; b) gauche; c) semieclipsed; d) staggered.

tical description of the silicon atom, and a third with the d functions being alloted to the chlorine. The fourth case corresponded to d character being allowed to both the silicon and the chlorine atoms. The usual kind of numerical data reported from SCF calculations are presented for this molecule in Table 1, from which it should be noted that the greatest stabilization upon adding d character to the molecule results from placing the d function on the silicon rather than on the chlorine. This stabilization is reflected in both the Si--Cl and Si-H overlap charges.The Mulliken charge on the silicon atom is particularly affected by putting the d orbital on the silicon, going from the no-d value of f0.3 to -0.3 upon adding d character to the silicon in this basis set. These findings would customarily be interpreted to show that, upon allowing d character to the silicon, there is a large transfer of charge to that atom; whereas, when it is allowed to the

Table 1. Some properties calculated for chlorosilane (H3SiCI). Property

Total energy [a.u.] Binding energy [a] [eV] Charge on Si Charge on CI Charge on H S-CI overlap S-H overlap

without d orbitals

- 749.483

+ -

9.0 0.31 0.12 0.06 0.61 0.69

silicon

-149.571 11.6 - 0.30 0.04 0.09 0.86 0.81

+ +

with d orbitals for chlorine

- 749.503

+ -

9.6 0.41 0.18 0.08 0.65 0.70

silicon + chlorine - 749.589 -

+

11.9 0.21 0.03 0.08 0.89 0.81

[a] The binding energy is the difference between the sum of the total energies of the constituent atoms and the total molecular energy, with 1 a.u.=27.21 eV and no correction for electron correlation.

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antibonding. Orbitals 3e and 4e are characterized by Si