Compton Profile, (e,2e) Studies, and “Atoms in Molecules”

N. L. ALLAN and N. H. MARCH Theoretical Chemistry Department, University of Oxford, I South Parks Road, Oxford

OX1 3TG, Englond

Abstract By direct comparison of ( c . 2 ) measurements with Compton lineshape determinations, we first

elucidate whether the ( e , 2 ) measurement strongly perturbs the molecular system. Specifically, results are presented for Nz and HzO. Although the e m r bars are somewhat large, excellent agreement is found between the two types of experiment. In order to relate such momentum space information 00 chemical bonding to other properties of interest, we have then examined: (i) The first moment of the measured molecular Compton profile. We show from the Thomas-Fermi statistical theory for molecules that this moment yields an approximation to the Dhc-Slater exchange energy, and values are given for N2 and HzO. (ii) The asymptotic form far from nuclei of the r space electron density which follows from density functional theory, and the justification it gives for “atomic-like building blocks” at large r or equivalently small momentum p. As predicted by the theory, the small p behavior of the Compton profile can be related between the equilibrium molecule, when the diffraction factor is removed, and the united atom, the link being established via the appropriate ionization potentials in the two configurations.

1. Introduction

A good deal of attention has been given to the study of Compton profiles in condensed matter, and most especially in crystalline solids. Interest in this field stemmed not only from the comparison between measured Compton lineshapes and energy band predictions, an early example, reviewed in Ref. 1, being the work of one of us [2 ,3] on crystalline lithium, but also from the hope of extracting significant information about electron correlation effects [4].

Progress has been somewhat slower in the case of measurements in the gaseous phase, but incentive for further theoretical work in this area has come from the progress on (e.2e) studies of the momentum space properties of valence electrons in molecules, an interesting example selected from many being the work on acetylene [ 5 ] . A further reason has been the interest in the properties of low- dimensional solids, which are often fruitfully discussed as the limit of large molecules as the number of chemically identical units tends to infinity. Examples which can be cited in this context are the one-dimensional case of the linear polyenes, or the quasi-two-dimensional examples of graphite and boron nitride, the basic molecular building blocks in these layer compounds being benzene and borazole, respectively [6].

This then was the motivation behind the present work. While it is still in its

International Journal of @ n N m Chemistry: Quantum Chemistry Symposium 17, 227-240 (1983) 8 1983 by John Wiley & Sons, Inc. CCC 0360-8832/83/010227-14504.00

228 ALLAN AND MARCH

early stages, it is of interest to report results on some admittedly rather well understood simple molecules, N2 and H20. However, we shall do so with two points in mind. First of all, the fruitful (e,2e) studies of valence electrons might nevertheless not be reflecting quantitatively the momentum space properties of the unperturbed molecule, but the act of making (e,2e) measurements might heavily perturb the system (cf. the use of the positron annihilation technique to study conduction electrons in simple metals, where massive pile-up of electrons round the positron* is known to occur; see, for example, the review by Jones and March [ 7 ] ) . Instead of attempting further theoretical analysis beyond existing treatments, we have chosen at this stage to bring measured (e,2e) momentum distributions into contact with Compton lineshape measurements. The conclusion is pretty encouraging for the (e,2e) technique; we find little evidence for strong distortion of the momentum space properties of the “unperturbed molecular system” by the act of making the (e,2e) measurements.

The second area explored, in the light of the above agreement between (e,2e) and Compton measurements, is the information contained in the low-order mo- ments of the Compton profile or the (e,2e) data. Here, we found that, at present, the Compton profile data have a large advantage; the high-momentum compo- nents in the (e,2e) studies are not sufficiently well determined to allow the computation of these moments to useful accuracy. From Compton data, however, the moments are known and we show how, with some assumptions, the first moment can be usefully related to the DiracSlater exchange energy in the molecule.

With this introduction, we turn immediately to compare (e,2e) and Compton profile measurements for two molecules, H20 and N2, for which both types of data are available.

2. Comparison of Compton Profile and (e,2e) Data

We briefly summarize the quantities that are measured in (a) Compton line studies and (b) (e ,2e) data.

A. Measured Quantities Related to Momentum Distribution

The Compton line has an intensity which can be expressed as a function of the deviation Ah from the Compton wavelength A, in the dimensionless form [ I ] :

= 27~ -( k p(k) dk.

*In metals, positrons rapidly thermalize; this is to be contrasted with the high energy of the incident electron in (e,2e) experiments.

(e.2e) MEASUREMENTS 229

Here I@)dp is the probability of an electron having momentum of magnitude between p and p + dp. p (p ) is the spherical average of the momentum density p(p) where this is, of course, related to the square of the momentum space wave function in the usual way. Differentiating Eq. (2.1) with respect to q yields

As to the (e,2e) experiment, one measures the distribution of momenta and energies of two electrons in coincidence after ionization by a high-energy electron beam of known momentum incident on a gas target. This technique has the unique feature that it measures the momentum distribution of individual orbitals. At sufficiently high incident energies and large momentum transfer, the cross section in the (e,2e) experiment is given by

a J 4: (PI dcl. (2.3)

A brief discussion of this is given in Appendix A, while a fuller treatment can be found in Ref. 8.

In Eq. (2.3), &(p) denotes the momentum wave function of the orbital being investigated, from which the electron is ejected. Different p values can be in- vestigated by varying the azimuthal angle at which one of the electrons is de- tected. Equation (2.3) assumes the plane-wave impulse approximation and a Hartree-Fock description of the target state. In the symmetric nonplanar ge- ometry, which has been assumed, all factors in u but the one exhibited in Eq. (2.3) can be kept constant.

B . Experimental Data for H20

Equation (2.3) shows that the (e,2e) technique therefore measures p(p) for each separate orbital, while the x-ray technique allows the total momentum density to be found. To illustrate this, and to see (as outlined in Sec. 1 ) whether the Coulomb interaction between incident and target electrons significantly dis- torts the momentum distribution, we have determined p(p) and I ( p ) for H20 (91 from the measured Compton profile by numerical differentiation. Figure 1 shows p(p) vs. p , the crosses being the experimentally determined Compton scattering data [9]. Although there is substantial scatter, the data indicate a maximum in p(p) around 0.5 a.u.

In Figure 1 , these Compton data are compared directly with those from the (e,2e) experiment, these latter experimental data simply being used by adding together p(p) for the individual orbitals from the graphical representation of the data presented in Refs. 10 and 11. The (e,2e) experiment does not measure the profile of the core 1s electrons (the corresponding incident electron energy is too high), so the contribution of the 1s electrons to the Compton profile (only a small fraction of the whole at small p ) was subtracted from the x-ray data presented in Ref. 9, using the values tabulated in Ref. 12.

1 -

(e.2e) MEASUREMENTS

x

Figure 2. Same as Figure 1 except that I (p ) [ = 4mp2p(p)] is plotted for H20.

23 1

Figure 3. Momentum density p(p) vs. p for N2. ( X ) Compton data [13]; (. . .) ( e . 2 ) data [15]. Theoretical Compton data in Ref. 14 were in satisfactory accord with the data.

232 ALLAN AND MARCH

x

x

Figure 4. I ( p ) = 4np2p(p) for N2. Otherwise same as in Figure 3.

3. Moments of Compton Profile and their Interpretation

Having established the essential equivalence of the spherically averaged mo- mentum distribution obtained from (e,2e) and Compton profile studies, we want to take up the question of the moments of the Compton lineshape and how they are to be interpreted.

As Coulson emphasized, the total kinetic energy T of the molecular electrons can be obtained from the second moment of the Compton profile:

where N is the total number of electrons, while J ( q ) represents the Compton lineshape in the usual reduced units.

The generalization of the above result for the mth moment is easily obtained as

and for atoms, since J ( q ) falls off at large q as an inverse power of q. it is evident that there are only a small number of finite moments for 0 S rn S 4. The bounded moments of p"' for negative p are p-' and p-'.

( e . 2 ) MEASUREMENTS 233

In 1950, Coulson and March [ 161 used the statistical theory of atoms to show that for atomic number 2 the mean momentum is given by

( p ) = 0.69 P3 a.u., (3.3)

a result which must be true in nonrelativistic theory in the limit of really large 2. Independently, Scott [ 171 calculated the Dirac-Slater exchange energy with the same Thomas-Fermi density to find

A = -C I ~ ( r ) " ~ dr = -0.221 Fa a.u. (3.4)

It is straightforward from the statistical theory to show that p and A are related by

as has been emphasized by Pathak and Gadre [ 181. They then showed, beyond statistical theory, that numerical calculations of H e - F o c k quality on atoms satisfy Eq. (3.5) to satisfactory accuracy.

It can be shown for homonuclear diatomics like N2 that the statistical theory again leads to Eq. (3.5). The allowed phase space is now more complicated geometrically than for atoms, and therefore we set out the statistical argument in some detail in Appendix B.

A. Mean Momentum from Measured Compton Profiles for N2 and H20

We have calculated, for the reasons outlined above, the measured mean mo- ments from the Compton profile data for N2 and H20. The values we obtained by numerical integration are

and ( P ) ~ ~ = 2.8 a.u./electron

( P ) " ~ = 2.7 a.u./electron.

These values allow the estimation of the Dirac-Slater exchange energy at the equilibrium internuclear separation from Eq. (3.5). We find the values

and AN? = - 12.0 a.u.

AHa = -8.6 a.u.

These values, derived from experimental Compton profiles are referred to again, briefly, in Sec. 5 , in relation to the dependence of A on the total number of electrons in the molecule.*

*Values for F2 are worth adding: (P)F* = 3.3, AF* = - 18.9

234 ALLAN AND MARCH

4. Asymptotic Form of r Space Density, Building Blocks, and Small p Behavior of the Compton Profile

We turn to the r space electron density, far from the nuclei. Some precise things can be said about this from density functional theory, and it should occasion no surprise that such precise knowledge at large r allows predictions to be made in momentum space at small p. The point we shall be emphasizing is the importance of characterizing p space data, at small p, by the molecular ionization potential I(R), where for homonuclear diatomics we have shown ex- plicitly its dependence on internuclear distance R. Of course, from experimental measurements, we have only information on the equilibrium value of R, say Re. Nonetheless, we can at least make some contact between the small p behavior of the homonuclear diatoms at small p, at R = Re, and the united atom mo- mentum distribution corresponding to R = 0.

Turning then, briefly, to the Euler equation of the density functional theory, we can regard it as expressing the constancy of the chemical potential p through the charge cloud:

March and Bader [ 191 pointed out that, at sufficiently large r, the kinetic energy T term can be evaluated exactly in terms of the ground-state electron density p(r), resulting in the Euler equation far from all nuclei as

1 v2p 1 (Vp)Z SU, + v,, + v, + -, W) p = - - - + - -

4 P 8 P 2 (4.2)

where U, is the correlation potential energy, V,, + V, being the Hartree potential energy.

But at sufficiently large r , the spherical average of the homonuclear electron density has the form:

(4.3)

in terms of the molecular ionization potential I(R). As Alonso and March [20] have pointed out, Eq. (4.2) can be solved asymp-

totically far from all nuclei for the square root of the density p"' = x, because then Eq. (4.2) becomes a Schrainger equation. Provided we prepare atomic- like building blocks for binding such that they are characterized by exponential decay with the molecular ionization potential inserted into Eq. (4.3), we then find

(4.4)

where the building block densities are taken to be centered on the nuclei A and B. This is strongly reminiscent of LCAO theory, although Eq. (4.4) is proved only asymptotically, far from nuclei.

p(r) - A? exp [ - 2 (21)'/2r],

p1l2(r) = p,!f(r) + pk2(r),

(e.2e) MEASUREMENTS 235

Then the density matrix in r space, y(r,r’), factorizes into the form at large r and r’:

y(r,r‘) = p112(r) p112(r’). (4.5)

A. Momentum Space Density Matrix at Small Values of Both Momenta

We next note that from the factorizable form (4.5) of the r space density matrix at large values of both its arguments, we can calculate the small p behavior of the momentum space density matrix y(p,p’) defined by

y(p,p’) = I y(r,r‘) exp (-ip * r) exp (ip‘ * r’) dr dr‘. (4.6)

Evidently, this factorizes to read, at small p and p‘

Y(P,P‘) = p’l2(r) P’/2(rf), (4.7) and so the small p behavior of the diagonal element p(p) y(p,p) is determined by the behavior of the building blocks at large r .

The important point to emphasize for our present purposes is that, when the transform is carried out, the essential variable in the description is seen to be p2/2ml(R). There is, of course, a diffraction factor of the spherically averaged form [ 1 + jo (pR)] due to the building block model.

B . Relation of Momentum Distribution for Homonuclear Diatoms to that of the United Atom

We have tested the above predictions on N l , with Si as the united atom, and on H2 and He. The former case is handled by taking the momentum distribution of Si from atomic calculations, this being shown in Figure 5. Then we removed the diffraction factor given above from the measurements for N2 [13] by numerical differentiation, the final results also being presented in Figure 5 . It should be noted that we have used the variable p/[2f(R)]1’2, I(R) for N2 being taken [21] from experiment as 0.573 a.u. The Si ionization potential was also taken from experiment, its value being 0.299 a.u., as in Moore’s tables.

It can be seen from Figure 5 that, although the overall shapes for N2 and the united atom are different, out to ~ / ( 2 1 ) ’ / ~ = 0.6 a.u., the small p behavior is similar for N2 and Si and, in fact, by multiplying by a constant, the two curves have been made to coincide for small p .

Of course, from the theoretical argument above, such agreement is only to be anticipated for small p , but in this context we have carried out similar cal- culations for H2 and the united atom He [23], and in this case there is agreement in terms of the scaled variable over a wide range of p , as shown in Figure 6. However, we expect this case to be exceptional.

In summary, it seems well worthwhile to use the scaled variable P / [ ~ ( R ) ] ” ~ , which we expect to bring molecular momentum distributions into coincidence

236 ALLAN AND MARCH

Figure 5 . Comparison of molecular I ( p ) for N2 divided by diffraction factor, and I ( p ) for the united Si atom. Independent variable is taken as p/(U)'R in a.u. Si curve ( . . . ) has been scaled by an arbitrary constant and then lies on top of the Nz curve ( X X x ), within experimental error, over a range of p from 0 to 0.6.

Figure 6. Same as for Figure 5 , but for H2 ( X ) and He ( . . ) united atom.

( e . 2 ) MEASUREMENTS 237

at small p for different values of the internuclear separation R, the united atom limit discussed above being a stringent test of the proposed scaling.

5. Summary

The main achievements of this article are as follows. (i) Although the (e,2e) experiment has the major merit that it yields information

about momenta in individual orbitals, we have compared the total momentum density for N2 and H 2 0 with experimental Compton data. While there are some problems in handling the (e,2e) data, the agreement is already encouraging. There is no evidence that the (e,2e) measurement is making a major perturbation of the molecular system in these two molecules, at the high incident energies involved in the experiments we have analyzed.

(ii) The mean momentum is shown, from the Thomas-Fermi statistical theory of molecules, to be directly related to the Dirac-Slater exchange energy. For the molecules on which we have found data (N2 and H 2 0 from experiment, F2 [24] and SiH4 [25] from theory), the exchange energy is found from the numerical estimates, over an admittedly small range of N, to be proportional approximately to Iv5’3, N being the total number of electrons [cf. the result due to Scott for neutral atoms in Eq. (3.4)].

(iii) The building blocks of an “atoms in molecules” method have been ex- amined. These have to take account of the molecular ionization potential. Then the “atoms in molecules” method has merit at small p for N2 and the united atom Si, and for a wider range of momenta for H2 and its united atom He.

In view of these encouraging results, a wider group of molecules than con- sidered here is currently being studied, using both experimental data and self- consistent-field results, when available.

Appendix A: Theoretical Background to (e,2e) Experiment

As outlined in Ref. 8, the cross section in the (e,2e) experiment at high enough energy and momentum transfer is given by

pA and pB being the momenta of the outgoing electrons, whilep, is the momentum of the incident electron. The Mott electron4ectron scattering factor is denoted by fee while (p@-’ I #) is the momentum representation of the overlap between the target and ion eigenstates. The average is over initial degeneracies while the sum is over final degeneracies.

If one makes the Born-oppenheimer approximation, then

238 ALLAN AND MARCH

where the cross section is proportional to the rotationally and vibrationally av- eraged square of the overlap function. Generally the vibrational average is dealt with by calculating the overlap at the equilibrium nuclear position. n,, the number of equivalent electrons in representation r, comes from the sum over degenerate final states. In the Hartree-Fock approximation, applied to the target, this reduces to

where &(p), as in the text, is the momentum wave function for the single- particle orbital c from which an electron has been removed, while lrL12 is the spectroscopic factor or pole strength. The important point to be made is that in the symmetric experiment for fixed 8, all the terms in Eq. (A.3) except the last factor are essentially independent of the azimuthal angle 4, leading to Eq. (2.3) in the text.

Appendix B: Statistical Theory Relating Average Momentum ( p ) to DiracSlater Exchange Energy for Molecules

Let us briefly summarize the argument relating the mean momentum (p) to the Dirac-Slater exchange energy for atoms, before turning to the proof for molecules. The statistical relation (ti = 1)

(B. 1)

where p,(r) is the maximum momentum at r, is spherical, and allows a quantity r (p) to be extracted for a given p(r). Then the relation analogous to Eq. (B.l) in p-space is [ 181

(B .a The generalization of Eq. (B.2) for molecules, where p(r) is no longer spher-

ical, is readily obtained by noting from Eq. (B. 1) that each surface of constant electron density defines from Eq. (B.l) a value of p. In the statistical theory, therefore, there is a unique relation between p and the volume enclosed by the corresponding constant electron density surface, say f l ( p ) . The spherically av- eraged momentum density p(p) in the molecule is then given by the generalization of Eq. (B.2) to read

p(r) = (1/3n2) p j ( r ) ,

P(P) = (1/3.rr2) fw.

P(P) = (l/4n3) fw, (B .3) reducing to Eq. (B.2), as it must, in the spherically symmetric case when n ( p ) =

413 n?(p). Obviously the mean momentum per electron is given by

1 P l

where N is the total number of electrons in the molecule.

(e,2e) MEASUREMENTS 239

The exchange energy in the DiracSlater approximation is 113

A = -c, p4'3 dr: c, = 4 t? (i) But we now use Eq. (B.1) to write

(B.5)

and since we have seen above that surfaces of constant p are characterized by their enclosed volume in r space, n(p), we can rewrite Eq. (B.6) as

Integrating by one finds

parts, and noting that one has no contributions from the limits,

A = +- " /i" Wp3(R) dp. (3P2)4/3

One now utilizes Eq. (B.3) to find

which in a.u. yields the same result [Eq. (4.3)] as for atoms. This relation [Eq. (B.9)] has been employed in the text to calculate A for H20 and NZ, using Compton profile data for (p).

Bibliography B. Williams, Compron Scattering (McGraw-Hill, New Yo&, 1977). N. H. March, Roc. Phys. SOC. 67, 9 (1954). B. Donovan and N. H. March, Roc. Phys. Soc. 69, 1249 (1956). G. E. Kilby, Roc. Phys. SOC. 82, 900 (1%3). M. A. Coplan, J. H. Moore, and J. A. Tossell, 1. Chem. Phys. 68,329 (1978); M. A. Coplan, J. H. Moore, J. A. Tossell, and A. Gupta, J. Chem. Phys. 71, 4005 (1979). See, for example, the review by one of us in Chapter 12 of Polymers, Liquid Crystals. and bw-DimensionaI Solids. N. H. March and M. P. Tosi, Eds. (Plenum, New York, 1983). W. Jones and N. H. March, Theoretical Solid-Srute Physics (Wiley-Interscience, London, 1973), Vol. I , p. 485. V. G. Neudachin, G. A. Novaskol'tseva, and Yu F. Smimov, Sov. phys. JETP 28,540 (1966); 1. E. McCarthy and E. Weigold, Phys. Rep. C 27,275 (1976); I. E. McCarthy, A. I. P. Conf. Roc. 86, 5 (1982). A. C. Tanner and I. R. Epstein, J. Chem. Phys. 61, 4251 (1974). S. T. Hood, A. Hamnett, and G. E. Brion, J. Electron Spectrosc. 11, 205 (1977). E. Weigold, A. J. Dixon, and 1. E. McCarthy. Chem. Phys. 21, 81 (1977).

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[12] R. J . Weiss, A. Harvey, and W. C. Phillips, Philos. Mag. 17, 241 (1968). [13] P. Eisenberger and W. A. Reed, Phys. Rev. A 5, 2085 (1972). [14) P. Eisenberger, W. H. Henneker, and P. E. Cade, J. Chem. Phys. 56, 1207 (1972). [IS] E. Weigold, S. Dey, A. J. Dixon, I. E. McCarthy, K. R. Lassey, and P. J. 0. Taubner, 1.

[16] C. A. Coulson and N. H. March, Roc. Phys. Soc. A 63, 367 (1950). (171 J . M. C. Scott, Philos. Mag. 43, 859 (1952). (181 R. K. Pathak and S. R. Cadre, J. Chem. Phys. 74, 5925 (1981); 77, 1073 (1982). [19] N. H. March and R. F. W. Bader, Phys. Lett. A 78, 242 (1980). [20] J. A. Alonso and N. H. March, J. Chem. Phys. in press (1983). [21] The vertical molecular ionization potentials were estimated from D. W. Turner, Molecular

(221 F. Biggs, L. B. Mendelsohn, and J. B. Mann, At. Data Nucl. Data Tables 16, 201 (1975). I231 P. Eisenberger, Phys. Rev. A 2, 1678 (1970). [24] L. C. Snyder and T. A. Weber, J. Chem. Phys. 68, 2974, (1978). [25] R. A. Ballinger and N. H. March, Proc. Cambridge Philos. SOC. 51, 504 (1955).

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Received May 5 , 1983