J . Phys. Chem. 1985, 89, 969-974 a phase space constraint, the procedure yields those same quantities corresponding to a nunconverged variational calculation and allows one to estimate the divergence error above Ed. For example, for a basis set defined by E < E,,,, the appropriate formula is a modification of eq 4.1 of ref 39
It differs from the ref 39 formula by the inclusion of a Maslov index 6 and the second step function O(E,,-Ho), which serves to restrict the available phase space to that enclosed by eq 4. Here, n is the dimensionality and N is an integer label which for n = 1 serves as the quantum number v. Equation A1 was applied to the Figure 2 example by using 6 = 1/2 and E,, = 10.5, and the resulting E(v) is shown as the solid line. Agreement with the variational eigenvalues is reasonably good. Appendix B. The “Orbit” Method in Two Dimensions
W e examine the Hamiltonian studied by Davis and Heller2’ 1 H = -(p? p,? w?x2 w,?y2) XxZy (Bl) 2 where wx = 1.1, my = 1.0, and X = -0.042. They obtained a semiclassical wave function for the (30,30) state by using a quantizing classical trajectory, and they stated that an accurate quantum-mechanical wave function would be “essentially impossible to generate” variationally. We were curious to find out yhether the orbit method, which enables us to choose a minimum Hobasis set, could in fact make such a calculation possible.
+
+
+
+
(39) Noid, D. W.; Koszykowski, M. L.; Tabor, M.; Marcus, R. A. J . Chem. Phys. 1980, 72, 6169.
969
To perform the analysis, we ran Davis and Heller’s quantizing trajectory. The separable Hamiltonian Howas chosen by removing the Xx’y coupling term from eq B1. The instantaneous values of position and momentum define a pair of instantaneous separable which in turn define a pair of instantaneous energies EOx,EOy, quantum numbers vx,vyof the corresponding classical Hostate. These quantum numbers form a path which sweeps out an area in vx,vyspace as the trajectory is propagated. The basis functions required to semiclassically convekge I+) are given by the integer-valued grid points within that area. Furthermore, assuming that the trajectory points are comptited at equal time increments, the density of points in a region of quantum number space is proportional to the classical overlap between I+) and a basis function 14) located in that region. The vx,vypath is shown in Figgre 5, which represents 1000 trajectory points. It is seen that this path, like the trajectory’s path in coordinate space, forms a regular, quasi-periodic pattern, with sharply defined caustics. The area of vx,vyspace covered encompasses roughly 600 in;eger grid points of A symmetry (even vx);therefore, about 600 Ho basis functions would be required to semiclassically converge this state. The actual number needed for an accurate variational calculation would be somewhat larger and would depend on the magnitude of quantum effects and 2n the accuracy desired. On the o t p r hand, a better choice of Ho may exist for which even fewer basis functions would be required. To perform the calculation, a reahonable procedure is to start with the vx,vybasis functions required for semiclassical convergence and then add on a surrounding border 6f basis functions; the border could be thickened until desired convergence was achieved. It seems probable that approximately 1000 or fewer basis functions would be sufficient, a calculation which is tractable on a large computer, althoug‘r beyond the resources available to us at the time of preparation of this paper.
Molecular Electron Density Distributions in Position and Momentum Space Diane C. Rawlings and Ernest R. Davidson* Department of Chemistry, University of Washington, Seattle, Washington 981 95 (Received: October 1 , 1984)
Comparison of molecular electron densities in position and momentum space provides some new insight into the interpretation of electron distributions in momentum space. Plots are given for the isoelectronic rholecular pairs (CsH5-, C,NH,), (N2, C2H2),and (C2H4,CH20) and for the array Hd9+arranged in a 7 X 7 two-dimensional square lattice.
Molecular electron density plots in position space provide a visual description of the electronic structure of molecules. These plots are easy to interpret because of their conformity with our usual physical perceptions. In momentum space such plots provide a different and very unfamiliar glimpse of the same electronic structure. Even though studies of atomic and molecular momentum wave functions have appeared in the literature since 1941,’” the difficulty of interpretation has prevented this viewpoint (1) Coulson, C. A.; Duncanson, W. E. Proc. Cambridge Philos. Soc. 1941,
37, 55, 67, 74, 397, 406; 1942, 38, 100, 1943, 39, 180; Proc. Phys. Soc., London 1945, 57, 190; 1948, 60, 175. (2) McWeeny, R.;Coulson, C. A. Proc. Phys. Soc., London, Sect. A 1949, 62, 509. (3) McWeeny, R. Proc. Phys. Soc., London Sect. A 1949, 62, 519. (4) Henneker, W. H.; Cade, P. E. Chem. Phys. Lett. 1968, 2, 575. (5) Weiss, R.J.; Harvey, A.; Phillips, W. C. Philos. Mag. 1968, 27, 241. (6) Epstein, I. R.;Lipscomb, W. N. J . Chem. Phys. 1970, 53, 4418. (7) Epstein, I. R. J . Chem. Phys. 1970, 53, 4425.
0022-3654/85/2089-0969$01.50/0
from becoming popular with chemists. In this paper molecular electron density plots in position and momentum space are e]rhibited for three isoelectronic pairs of molecules. Cyclopentadietle anion and pyrrole differ by replacing C- by N. Acetylene and p2differ by replacing two C-H groups with two nitrogens. Ethylene and formaldehyde differ by replacing a CH2 group by an oxygen atom. These examples allow comparison of the densities in,very similar molecules. An orbital in momentum Bpace, *(k), is given by the Fourier transform of the position space orbital, 4(r) Q(k) = ( 2 ~ ) - ~ / ’ 1 4 ( rexp(-ikmr) ) d7
(1)
It is important to note that this transformation preserves directions so that it is still correct in momentum space to talk about directions perpendicular to the plane of a planar molecule and (8) Cook, J. P. D.; Brion, C. E. Am. Inst.Phys., Conf. Proc. 1982,86,278.
0 1985 American Chemical Society
970 The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
Rawlings and Davidson
Figure 3. Valence electron density of pyrrole in position space. Oriented with nitrogen on the left.
marked by a " f "
Figure 4. Valence electron density of pyrrole in momentum space. Oriented as in Figure 3. Figure 2. Valence electron density in the plane of the molecule for cyclopentadiene anion in momentum space. Straight lines mark symmetry unique bond directions and directions perpendicular to bonds.
directions parallel to certain bonds. There are important distinctions, however. The vector joining two atoms in position space has a well-defined direction, length, and origin. In momentum space only the direction is preserved and information about the origin is lost. In certain cases information about the length, R, may appear as oscillations in the density with wavelength 2?r/R. For a real space orbital, the complex conjugate of CP(k) is *(-IC). Therefore, the momentum orbital can be expressed in terms of a real part Re CP = '/*[CP(k)
+ CP(-k)]
(2)
- CP(-k)]
(3)
and an imaginary part 1
Im 3 = -[3(k)
2i
Adding the square of the real and imaginary parts, one obtains the identity p(k) = p(-k) = 3(k) CP(-k)
(4)
Therefore, as mentioned by Lipscomb and Epstein,6 the momentum density will exhibit inversion symmetry. This can be seen in the plots of the valence densities of cyclopentadiene anion and
Figure 5. Nitrogen s-s overlap density between two centers 1.0975 A
apart. pyrrole in Figures 1-6. Although in position space pyrrole has C, symmetry, the transformation to momentum space introduces inversion symmetry leading to D2*symmetry in the momentum density. The cyclopentadiene anion is D5,,in position space but
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985 971
Molecular Electron Density Distributions
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TABLE I: Basis Set and Geometry Information molecule basis set geometry N2 DZP R(N-N) = 1.0975 8, C2H2 DZP R(C-C) = 1.2070 8, R(C-H) = 1.0580 8, C2H4 DZP R(C-C) = 1.3300 8, R(C-H) = 1.0796 8, LHCC = 121.7O CHiO DZP R(C-0) = 1.2099 8, R(C-H) = 1.1199 8, LHCO = 121.0° C4NH5 SVb R(N-C1) = 1.3703 & R(C1-C2) = 1.3820 8, R(C2-C3) = 1.4168 8, LCNC = 107.7O C5H5SVb R(C-C) = 1.3933 8, R(C-H) = 1.0801 8,
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Figure 6. Nitrogen p p u overlap density.
the momentum density has Dlohsymmetry. Similarly, the real part of O(k) is of gerade symmetry and the imaginary part is of ungerade symmetry. For real orbitals, @(a) is the Fourier transform of 4(-r). Hence, one can also see from eq 2 and 3 that for orbitals which are gerade in position space, Le., when $(r) = +(-r), the imaginary part of O(k) will be zero. Similarly for ungerade orbitals, Le., when +(r) = 4(-r), the real part of O(k) will be zero. O(k) will also reflect the other symmetry properties of $(r). For example, if the point group operator, T,changes $(r) into $(T’r) = X4(r), then T 1changes O(k) into XO(k). One consequence of this is that an orbital 4(r) of a’’ symmetry under reflection will give a O(k) of a’’ symmetry. Thus in Figures 1-4 only the a’ orbitals contribute. Geometry and basis set information is collected in Table I. Total overlap density plots were generated by summing the SCF occupied orbital densities for the molecule and subtracting the atomic orbital densities multiplied by their respective gross Mulliken o c c ~ p a t i o n s mjA: ,~
where the sums on Z and j run over molecular and atomic orbitals, respectively. The sum on A runs over nuclei. Mulliken gross populations were used so that all of the atomic-like density could be removed without making an arbitrary assignment of overlap populations to atomic orbitals. Previously published papers on momentum distribution^'.^ have commented that the momentum for diatomics such as Hz is higher perpendicular to bond directions and lower along bond directions. Epstein and Lipscomb found that density maps of BzH6 and trans-butadiene did not show this pattern, and they attributed it to the obscuring effects of overlapping contributions from various bonds.6 We find that only for the case of u bonds that have significant s character is there increased momentum perpendicular to the bond and decreased momentum parallel to the bond. For u bonds which are sums of p functions the situation may be reversed: momentum is increased along the bond and decreased perpendicular to the bond. Figure 5 shows the plot of the overlap density coming from a bond between s orbitals on two centers & I s s= SA*SB SB*SA (6)
‘Dunning [3s,2p,ld] contraction of Huzinaga’s (9s,5p) for C and N. Our own [3s] contraction of (5s) for H. bDunning [3s,2p] split-valence contraction of Huzinaga’s (9s,5p) on C and N and a [2s] contraction of Huzinaga’s (4s) on H. cL. Nygaard, J. T. Nielson, J. Kirchheiner, G. Malteson, J. R. Anderson and G . 0. Sorenson, J. Mol. Struct., 3, 491 (1969). the region of low momentum because the cosine is positive near zero momentum. It will be noted that, while large R leads to a small overlap density in position space, the amplitude in momentum space does not decrease with R. In fact, the overlap density and the atomic density have comparable magnitudes for all R . In momentum space large R is characterized by rapid oscillations in the overlap density. The plot in Figure 6 shows the overlap density from a bond between p u orbitals on two centers (both oriented with positive lobes to the right and with center B to the right of center A)
This shows a region of increased momentum along the bond direction just as in the actual N2 result. The nodal pattern here results from the product of the p orbital density with a node through zero momentum and the oscillations of the cosine function. App is negative in the region of low momentum because of the overall minus sign in eq 9 compared with eq 7. Similarly, the overlap density from a bond between an s orbital on center A and a p orbital on center B takes the form:
po(k) is pure imaginary so ipo is real. Further, isopo is positive for positive k, so Ap is positive near zero momentum. It is also interesting to note the mirror image bond
gives the same overlap momentum density. This formula also shows that the one center sp hybrid orbital density (the R = 0 limit) vanishes in momentum space. These figures may be combined to obtain the overlap density associated with a bond between sp hybrid orbitals on each center:
+
=
~ s O * S O COS
[k*R]
(7)
where so(k) is the momentum orbital for an s orbital at the origin in position space and R = B-A. As noted by Cook and Brion, the nodal pattern results from the oscillations of the cosine function which is periodic with a period 2rlR. The result is positive in (9) Mulliken, R. S. J . Chem. Phys. 1955, 23, 1833, 1841, 2338, 2343.
This overlap density makes a large contribution in the region of low momentum. Total density difference plots are presented in position and momentum space for the two isoelectronic molecules, Nz and acetylene (C2H2),in Figures 7-10. Both plots are quite different from previously published plots of N, by Henneker and Cade4 and acetylene by Epstein.’ Epstein’s plot was base on subtraction
972
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
i
Rawlings and Davidson
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Figure 7. Total density difference plot of N2 in position space.
Figure 10. Total density difference plot of acetylene in momentum space.
Figure 8. Total density difference plot of N2 in momentum space.
Figure 11. Total overlap density plot of ethylene in position space parallel to the plane of the molecule.
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Figure 9. Total density difference plot of acetylene in position space.
of spherically averaged s2p2carbon atoms while ours is based on the Mulliken gross population, so6sp283r,0 71r:7 1 with so32 on H. The N2 plots by Henneker and Cade were based on subtraction of N atoms in their valence state, s2p3. Ours is based on the N2 Mulliken gross population, si 69p6106r,o 74r: 74.
t
