Electron momentum spectroscopy of the valence orbitals of H2O and D2O: Quantitative comparisons using Hartree—Fock limit and correlated wavefunctions

19

Chemical Physics 113 (1987) 19-42 North-Holland, Amsterdam

ELECTRON MOMENTUM SPECTROSCOPY OF THE VALENCE ORBITALS OF H,O AND D,O: QUANTITATIVE COMPARISONS USING HARTREE-FOCK AND CORRELATED WAVEFUNCTIONS A.O. BAWAGAN, Department

Received

C.E. BRION

of Chemistry, The University ofBritish Columbia, 2036 Main Mall, Vancouver, B.C., Canada V6T 1 Y6

E.R. DAVIDSON Department

LIMIT

of Chemistry

24 November

and D. FELLER Indiana Uniuersity, Bloomingon, IN 47405, USA 1986

The large discrepancies found earlier between experimental measurements and calculations based on near Hartree-Fcck wavefunctions for the valence orbital electron momentum distributions of H,O are reinvestigated. New and improved electron momentum spectroscopy measurements for the valence orbitals of Hz0 and D,O, together with existing experimental data, have been placed on a common intensity scale using the binding energy spectra. Investigation of possible vibrational effects by means of new measurements of the momentum distributions of D,O indicates no detectable differences with the H,O results, within experimental error. A quantitative comparison of these experimental results with both the shapes and magnitudes of momentum distributions calculated in the PWIA and THFA approximations using new, very precise Hartree-Fock (single-configuration) wavefunctions is made. These wavefunctions, which include considerable polarization and which are effectively converged at the HF limit for total energy, dipole moment and momentum distribution permit establishment of basis set independence. The significant discrepancies between theory and experiment which still remain for the momentum distributions of the lb,, 3a, and 2a, orbitals at the THFA level are largely removed by CI calculations of the full ion-neutral overlap amplitude. These CI wavefunctions for the final ion and neutral ground states, generated from the accurate HF limit basis sets, recover up to 88% of the correlation energy. The present work clearly shows the need for adequate consideration of electron correlation effects in describing the low-momentum parts of the lb,, 3a, and 2a, electron distributions, a region which is of crucial importance in problems related to chemical bonding and reactivity. The high level of quantitative agreement obtained between experiment and calculations using sufficiently sophisticated wavefunctions provides support for the essential validity of the plane wave impulse approximation as used in the interpretation of EMS experiments on small molecules.

1. Introduction Electron momentum spectroscopy (EMS), also formerly known as binary (e,2e) spectroscopy [l-4], is a powerful emerging technique for the study of atomic and molecular orbitals and the laboratory investigation of molecular wavefunctions and chemical bonding [5,6]. Following its development over the past decade, the technique of EMS is now providing an increasing flow of experimental data which offers new possibilities in the area of quantum chemistry [4]. .As such, detailed comparisons between the results of EMS

experiments and state-of-the-art theoretical quantum chemical calculations should prove fruitful in further understanding and refinement of both experiment and theory. For gas phase molecules studied with high-energy electron beams, the EMS cross section within the plane wave impulse (PWIA) and target Hartree-Fock approximations (THFA) is expected [1,2] to be proportional to the spherically averaged momentum density of the orbital from which an electron is ejected. Such a relationship has already been demonstrated to be quite accurate for a number of atoms and small molecules

0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

20

A.O. Bawagan et ul. / EMS

ofthe valence orbitals of Hz0 and D,O

under appropriate conditions [l-4]. To clearly differentiate between the experimental results and theoretical calculations, EMS cross sections (as a function of (p) will be referred to as experimental momentum profiles (XMPs). Likewise further distinction is made between the different levels of theoretical treatment used in the interpretation of the observed XMPs. The calculated orbital momentum distributions within the THFA (see eq. (2) below) will be referred to as MDs whereas the (more accurate) ion-neutral overlap distributions (see eq. (1) below) will be referred to as OVDs. Several studies [7-lo] of the electron momentum spectroscopy of the valence orbitals of water’ have revealed certain unexpected features. Most notably the theoretical MDs calculated using what might normally be considered to be quite good wavefunctions showed significant discrepancies for the outer valence orbitals when compared with the XMPs. In particular all the studies [7-lo] ,indicated that the XMPs for the two outermost valence orbitals (lb, and 3aJ of H,O have significantly more density at low momentum than was predicted by THFA calculations of MDs using wavefunctions which were presumed to be near the Hartree-Fock limit. In addition the calculated peak in the MD occurred at a momentum, prnax which was higher than that observed experimentally for the lb, and 3a, orbitals. This observation, which in part suggests a greater spatial extension of the orbital than predicted by the THFA, has been discussed at length in a recently reported high-momentum resolution EMS study of H,O [lo] in which the discrepancy between calculation and experiment was found to be greatest for the outermost valence orbital (i.e. the non-bonding lb, orbital). It is of interest to note that similar discrepancies between theory and experiment have also been observed at the THFA level for the outermost valence orbitals of the neighboring second-row hydrides HF [ll] and NH, [12-141 although reasonable agreement exists for the isoelectronic species Ne [15] and apparently also for the third-row and higher analogues, PH, [16], H,S [17], HCl [ll], HBr and HI [18]. However it should be noted that the observed XMPs in the cases of H,S and PH, were obtained at poor momentum

resolution and the measurements need to be repeated at higher momentum resolution than was used in the preliminary work in order to confirm these findings. In addition, higher quality wavefunctions than those used earlier need to be employed in order to establish whether the generally better agreement observed to date for third-row hydrides is fortuitous *. In the recent paper on H,O [lo], the following considerations were raised as being possible sources of the observed discrepancies between the theoretical MDs and the XMPs. (1) Inadequacies in the plane wave impulse approximation (PWIA) and the need for distorted wave treatments (DWIA). (2) Inadequacies in the target Hartree-Fock approximation (THFA) and therefore the need to consider sufficiently complete target molecule-final ion overlap treatments (i.e. adequate treatment of relaxation and correlation). (3) Neglect of nuclear motion, i.e. vibrational effects. (4) Insufficient flexibility of the basis set and resultant deficiencies in the theoretical wavefunction. These matters concern both the adequacy of the theory of EMS cross sections and the accuracy of the wavefunctions. Explanations (l)-(3) emphasize possible inaccuracies in the assumed proportionality between the experimental momentum profiles and the orbital momentum distribution evaluated at the equilibrium geometry of the neutral molecule. Items (2) and (4) are of particular concern from the quantum chemical theoretical standpoint and these are investigated in detail in the present work. A further area of possible concern from an experimental standpoint is the accuracy with which the momentum resolution is known. Such effects however will be small at higher momentum resolution (as used in the present experimental work) compared to the observed discrepancies between the measured XMPs and the calculated MDs in the case of H,O.

* Very recent high-resolution

EMS work on H,S [19] confirms that good agreement between experiment and theory is achieved at the THFA level using HF limit wavefunctions.

A.O. Bawagun et al. / EMS of the valence orbitals of H_,O and DzO

The validity of the PWIA for the present experimental conditions (impact energy = 1200 eV, E,=Ez=6Q0 eV, 8=45”, p~2 a;‘) has been discussed in ref. [lo] and has been clearly demonstrated by the complete agreement between EMS experiment and theory for atomic hydrogen [20] where the, THFA is not involved and the wavefunction is exact. A further proof of the independent validity of the PWIA in the case of a nonpolar many-electron target was provided by the very close agreement with experiment obtained in the PWIA calculations of Mitroy et al. [21] using either the THFA or the full ion-neutral overlap treatment for the valence orbitals of argon. There is thus good reason to use the PWIA for other atomic and molecular targets under the equivalent conditions. The combined use of the PWIA for the kinematic factor together with use of the THFA for the electronic structure factor in the description and interpretation of the binary (e,2e) reaction as studied by EMS is also supported by the excellent results obtained for the valence orbitals of noble gas [15, 221 and metal atoms [23] and also for small molecules [6, 241. In these cases where high-quality Hartree-Fock limit wavefunctions were used it is clear that the combined use of these two approximations is valid since the shapes of the measured XMPs were extremely closely reproduced by the calculated orbital momentum distributions at least for electron momenta less than z 2 ai * for targets lighter than Ar (2 = 18) *. In the case of Ne (Z = lo), which is isoelectronic with H,O, the PWIA was satisfactory [15] in describing the shapes of the momentum distributions out to p = 2 a, I, In the region of higher electron momentum (i.e. larger than that used in the present work for H,O) considerable distortion of the electron waves can occur and the distorted wave impulse approximation (DWIA) has been found to be necessary for an adequate interpretation of EMS measurements for example in Ar [25],

* Experiments

[15,22] indicated that distortion becomes more important at lower momentum as the orbital energy and 2 increase. For example, even for targets as large as Xe (Z = 54) use of the PWIA was found [15,22] to be adequate for the valence orbitals out to 1.5 a;’ at 1200 eV impact energy.

21

Xe 1221, Ne and Kr [26]. In the region of p below = 2 a,’ the shape of the XMP for the noble gases was equally well reproduced by either the DWIA + THFA or PWIA + THFA. Thus far distorted wave treatments for molecules have not been reported due to the complex multicenter nature of the problem. In some molecules it has been found that THFA calculations using even Hartree-Fock limit wavefunctions fail to reproduce the experimental momentum profiles for some orbitals. This is most likely due to the inherent neglect of electron correlation and relaxation in the Hartree-Fock description. In such cases calculation of the full target ion-neutral overlap using CI wavefunctions can be used to include correlation and relaxation effects. An earlier attempt (generalized overlap amplitude (GOA) method) by Williams et al. [27] at evaluation of the target molecule-final ion state overlap for H,O [27], HF and HCl [ll] using limited basis set wavefunctions indicated small changes (towards lower momentum) from predictions based on ,the THFA. However more exhaustive treatments of overlap, using improved wavefunctions are needed to investigate more completely the effects of correlation and relaxation. From the standpoint of theoretical quantum chemistry it is important to assess the need for taking into account ground and/or final state electron correlation effects since these have certainly been found to be crucial in reconciling theory and EMS experiments in the case of NO [28] as well as for H, [24] and He [29] in those cases where the product ions of H, and He are left in excited states. As has been shown earlier by Weigold and co-workers [8,22] and discussed by McCarthy [30], additional quantitative assessment of the wavefunctions and the reaction model is possible if the experimental momentum profiles for the valence orbitals are obtained on a common (relative) intensity scale ** with the full Franck-Condon width of each final ion electronic state being taken into consideration in the normalization procedure. ** True absolute perform example

EMS measurements are very difficult to with high accuracy. Such measurements have for been reported ( f 20%) for He and H, [31],

22

A.O. Bawagan et al. / EMS of the valence orbitals of H,O and D,O

With a single point normalization of experiment and theory at only one value of the momentum on a given calculated orbital MD or OVD a stringent quantitative test of theory can be made at all other experimental data points and calculations for all orbitals. This procedure, which has also been used in the present work, provides very much more specific information than the individual height or area normalization of theory and experiment for each separate orbital which has frequently been used in earlier EMS studies. Flexibility of the basis set and the accuracy of the wavefunction are important concerns from the theoretical standpoint. The usual variational treatment involving energy optimization of SCF wavefunctions, even when energies near the HartreeFock energy are given, is a necessary but usually insufficient condition alone for guaranteeing that all other calculated properties are also close to the Hartree-Fock properties, as is well known [28,32]. The addition of more diffuse functions is usually necessary to permit accurate calculation of properties such as the dipole and quadrupole moments (and therefore also likely the MDs) which depend critically on the accurate modeling of the longrange charge distribution. The total energy is, of course, not very sensitive to such details so that basis sets designed solely to yield a reasonable value of the total energy without unnecessary computational expense may well give poor results for other properties. Further considerations in addition to the total energy are therefore necessary to ensure sufficient wavefunction accuracy for general applications where theoretical quantum chemistry is required to give accurate predictions of experimental observables. Such considerations are also of importance for a number of properties including those dependent on the long-range distribution of charge density. After convergence has been obtained at the Hartree-Fock level for both total energy and the predicted MD then the other possible sources of any remaining discrepancy between the calculated orbital MDs and the XMPs measured by EMS can be investigated. As mentioned above the need to go beyond the THFA (i.e. to incorporate correlation and relaxation effects) can be investigated with the use of suitable CI wavefunctions in order

to evaluate the ion-neutral overlap amplitude with sufficient accuracy. With the above considerations in mind we have made new higher statistical precision EMS measurements of the momentum profile for the lb, orbital of H,O as well as new momentum profile measurements for the three outer valence orbitals of D,O. These new measurements taken together with the earlier reported experimental data [lo] for the four valence orbitals of H,O have been placed on the same (relative) intensity scale and are now compared on a quantitative basis with THFA (MDs) and full overlap calculations (OVDs) using very accurate Hartree-Fock limit and CI wavefunctions, respectively. These include the already published 84-GTO wavefunction of Davidson and Feller [33] as well as the 39-ST0 wavefunction of Rosenberg and Shavitt [34]. These best existing literature wavefunctions [33,34] have been improved upon in the present work by use of previously unreported 99-GTO and 109-GTO wavefunctions for the THFA calculations. These new SCF wavefunctions for water, which give essentially converged results * for total energy, dipole moment and MD are considered to be the most accurate to date. Comparisons of theoretical MDs calculated from essentially converged Hartree-Fock quality wavefunctions are important in order to determine whether basis set independence of the calculated MDs has been achieved as well as energy minimization. Comparative studies of basis sets for the water molecule [33,34] have shown that certain position space one-electron properties such as the dipole and quadrupole moments are much more sensitive to the basis set than the total energy. Most theoretical efforts to date have been directed towards assessing the effect of “basis set truncation” on position space properties. There has in general been much less attention given to the conjugate momentum space properties which * These results are considered to be effectively converged since the total energy is within 0.5 mhartree of the estimated Hartree-Fock limit [34] and because the dipole moment, quadrupole moment and MD do not change significantly as even more basis functions are added. Other basis sets containing up to 140 GTOs were actually tested with no significant change in either the MD or OVD [35].

23

A.O. Bawagan et al. / EMS of the valence orbitals of Hz0 and DzO

emphasize a different region of phase space. Tanner and Epstein [36] have calculated the Compton profile and momentum expectation values for H,O using various SCF wavefunctions from minimal basis to near Hartree-Fock quality. It was observed that distinctions between Compton profiles calculated using wavefunctions of DZ quality or better (e.g. near Hartree-Fock) were extremely difficult. However this is not so surprising since the Compton profile refers to the total electron distribution rather than to an individual orbital. Leung and Brion [5,6] have investigated basis set effects on the momentum distribution of Hz(lu,J and found these to be small. However it should be noted that the “s-type” shape of the lus orbital of H, affords much less stringent assessment of the calculated MDs than do orbitals with MDs of “p-type” shape*. The maximum at nonzero momentum afforded by the “p-type” molecular orbitals in water therefore provides a sensitive probe of the basis set problem and the limitations of the Hartree-Fock model in momentum space. The most accurate H,O basis sets (i.e. the 99-GTO and 109-GTO wavefunctions, which are converged for both energy and THFA MDs) have been used in the present work as the foundations for more accurate treatments beyond the Hartree-Fock (independent particle) picture. To elucidate the effect of correlation and relaxation, a configuration interaction (CI) treatment (recovering a very large fraction of the correlation energy) of the initial and final states is used in a calculation of the ion-neutral overlap amplitude in the present work. The CI wavefunctions are generated using the highly extended basis sets and the calculated ion-neutral overlap distributions then compared with the experimental momentum profiles. The role of vibrational motion on momentum distributions has yet to be fully investigated but preliminary calculations for the symmetric stretch

* It should

be noted that a much greater specificity was observed by Leung and Brion [5] for calculations using different basis sets compared to the measured cross section when the difference density (pH, - 2p~) was used instead of the density.

in Hz0 [37] and for NH, [14] have indicated that the effects are very small for all valence orbitals and essentially negligible for the non-bonding lb, and 3a, orbitals, respectively. Isotopic effects have been investigated and found to be negligible in EMS experiments comparing the XMPs of H, and D, [38]. The new experimental data for deuterated water (D,O) therefore provide opportunity to investigate any effects of nuclear vibration on the momentum profiles. The different vibrational frequencies [39] of D,O and H,O may reveal within the experimental sensitivity any vibrationally induced effects. D,O due to its higher mass (thus lower vibrational frequency) would be expected to exhibit less of the vibrational effects. Consider for example the molecular root mean square vibrational amplitudes. The ((ARot+)2)1’2 for H,O is 20% greater than D,O and the ((ARmi)2)1/2 for H,O is 20% greater than D,O ]401. In summary then the present study seeks to address the various possible explanations for the observed discrepancies between experiment and theory in H,O. In turn this should provide insight into the nature of the outer valence orbital momentum distributions (i.e. XMPs and theoretical MDs) of other second-row hydrides (e.g. NH, and HF) which have been found to exhibit similar behavior. A brief discussion of the theoretical methods employed in the study is presented in section 2 followed by comments on the experimental details in section 3. The results are presented and discussed in section 4 and in addition, some ideas and proposals for further study are highlighted.

2. Theoretical methods 2.1. Theory of electron momentum

spectroscopy

Extensive discussions of the theory of electron momentum spectroscopy (i.e. binary (e,2e) spectroscopy) have been reported earlier [1,2]. The triple differential cross section, uTDcs for the (e,2e) reaction involving a randomly oriented N-electron system initially in its ground electronic state (@‘) and an (N - I)-electron ion in the final state

A.O. Bawagun et al. / EMS of the valence orhitals of H,O and D,O

24

( efN-‘) uTDCS

is given by =

THFA in the case of the water molecule by also performing accurate calculations of the ion-neutral overlap amplitude (see eq. (1)) using the new highly extended (i.e. converged) basis sets to generate CI wavefunctions for both the initial neutral state and the final ion state.

(4~3~l~2/~O)uMott

= constant X aMOtt

2.2. Extended In eq. (1)‘the plane wave impulse approximation [l] has been invoked and usually the vibrational integral is approximated by assuming the molecule to be at its equilibrium geometry. As will be shown later these approximations are very reasonable for the present experiments. Since the Mott cross section, uMvlott is effectively constant within the energy regime of the symmetric noncoplanar kinematics employed in the present EMS studies [15], the differential cross section is directly proportional to the spherical average of the square of the Fourier transform of the ion-neutral overlap. Previous studies have usually considered the independent particle approximations to this overlap whereby the dependence of uTDCS on the initial ground state and final ion states is reduced simply to the ionized orbital of the initial state of the target, i.e. (7 TDCS

=

amant

X

(IMott

/

dfi

I+,

(

P > 129

(2)

where #,(p) is the momentum space orbital wavefunction for the ionized electron. This approximation is valid if the initial state can be adequately described by a single Slater determinant of Hartree-Fock orbitals and the final ion states as linear combinations of similar determinants (i.e. using the same molecular orbital basis) coupled to a hole in the jth orbital [1,2]. In general, a single term ($,) results from the ion-neutral overlap and this has normally been referred to as the characteristic orbital. This in brief is the target Hartree-Fock approximation *. The present study investigates the validity of the * In the treatment of EMS measurements at the THFA level final state correlation may be included [25]. In most cases final state correlation is not included and the THFA the% reduces to the frozen orbital approximation (i.e. Koopmans theorem).

basis sets for H,O

Several new, highly extended basis sets for H,O have been developed in the course of the present work. First, a 99 basis function set (referred to as 99-GTO) was constructed from an even-tempered (19s,lOp,3d,lf/lOs,3p,2d) primitive set which was contracted to [lOs,8p,3d,lf/6s,3p,ld]. The s-components of the Cartesian d-functions were deleted, but the p-components of the Cartesian f-functions were retained. The (s, p) portion of the oxygen basis was created by extending the (18s,9p) atom optimized exponents with one additional set of diffuse functions of each type. The hydrogen s-exponents are optimal for the isolated atom. In an even-tempered basis [35] the i th exponent is given as ~$3’ beginning with i = 1. The (Y and j3 values for the oxygen (18s,9p) set are (Y, = 0.07029, & = 2.29663 and (Ye= 0.04956, & = 2.57217. The hydrogen values are (Y, = 0.02891, j3, = 2.58878. Exponents for the d- and f-type polarization functions on oxygen were partially optimized at the SD CI level. The d exponents are 3.43, 1.18 and 0.33. The f exponent is 1.20. For hydrogen the pand d-type polarization functions were taken from the work of Davidson and Feller [33]. The 109 basis function set (referred to as 109a (23s,12p,3d,lf/lOs,3p,2d) to GTO), [14s,lOp,3d,lf/6s,3p,ld] contraction, was generated by extending the (19s,lOp) set through the addition of more diffuse functions, while still retaining the even-tempered restriction. The most diffuse s function in this set possessed an exponent of 0.0649 compared to 0.1375 in the original set. While these basis sets, incidentally, give the lowest SCF energy yet reported for water at the experimental geometry, they are also designed to saturate the diffuse basis function limit and to give an improved representation of the (r-space) tail of the orbitals. The negligible difference between the 99-GTO and 109-GTO results for a

A.O. Bawagan et al. / EMS of the vulence orbit&

25

of H,O and D,O

Table 1 Properties of theoretical SCF and CI wavefunctions Wavefunction

Basis set

Energy (au)

Dipole moment (D)

(r”> (lo-l6

Ref. cm*)

RHF

SB [42/2] 14 CGTO

- 76.0035

2.681

5.382

1411

RHF

APC (331/21) 27 ST0

- 76.00468

2.035

5.462

[421

RHF

NM [531,‘21] 36 CGTO

- 76.044

2.092

5.307

1431

RHF

NM (1062/42) 58 CGTO

- 76.059

1.996

5.371

1431

RHF RHF

RS 39-ST0 DF 84-GTO 84 CGTO 99-GTO 99 CGTO 109-GTO 109 CGTO

- 76.0642 - 76.06661

1.995 2.021

5.396 5.417

[341 1331

- 76.06689

2.006

5.430

this work

- 76.0671

2.006

5.430

this work

140-GTO 140 CGTO

- 76.0673

1.981

5.432

1351

- 76.0675 a)

1.980 + 0.01 b,

5.432 + 0.001 b,

RHF RHF RHF Hartree-Fock

limit

CI CI

(84-GTO)CI (109-GTO)CI

- 76.3210 - 76.3761

1.929 1.895

5.490 5.500

this work this work

CI

(140-GTO)CI

- 76.3963

1.870

5.507

1351

- 76.4376 f 0.0004 a)

1.8546 f 0.0006 a*C

5.1* 0.7 a)

experimental

a) Ref. (341. b, Ref. [44]. ‘) Estimated non-vibrating dipole moment is 1.848 D.

wide variety of calculated properties (table 1) indicates that this goal has been essentially accomplished. The neutralOmolecule equilibrium geometry, R,, = 0.9572 A, t?nOH= 104.52” was used for generating all wavefunctions. CI calculations were then done for the 84-GTO and 109-GTO sets using configurations built from the neutral molecule SCF MOs. Based on these small CI calculations, the important configurations were selected and used in a multireference SD CI with perturbation energy selection of the configurations. For example, for the 109-GTO basis set, we used 15 configurations in the reference set and 11011 in the final CI for the neutral molecule X IA, wavefunction. Similarly we used 37 configurations in the reference set and 17316 in the final CI for the ion X 2B, wavefunction. These are still small CI calculations which recover only about 83% of the estimated correlation energy in

H,O. These CI wavefunctions were then used to evaluate the molecule-ion “overlap” (X ‘B, 1 X ‘A,), (X ‘A1 IX ‘A,), and (X *B, 1X ‘A,) which have the same form as an MO expanded in this basis. In order to generate the spherically averaged momentum distributions from each of the wavefunctions use has been made of HEMS, a comprehensive FORTRAN program developed at UBC. This program improves and integrates earlier separate programs for calculating spherically averaged MDs and density maps in both position and momentum space. Additional routines to handle f-type gaussian and Slater functions have been incorporated to allow the use of wavefunctions with very diffuse basis functions of the types used in the present work for the 84-GTO, 99-GTO, 109-GTO and 140-GTO wavefunctions.

26

A.O. Bawagan et al. / EMS of the valence orbitals of H,O and D20

3. Experimental method Details of the symmetric non-coplanar EMS spectrometer and its operation have been presented elsewhere [15]. In summary, following electron impact ionization (E, = 1200 eV + binding energy) the two outgoing electrons ( EA = E, = 600 eV) are detected (0, = 19, = 45 “) in coincidence as a function of $ (q - $ is the relative azimuthal angle). Binding energy spectra are generated at a given + (i.e. electron orbital momentum) by variation of E,. Alternatively orbital momentum distributions are obtained by variation of $J (0 to +30”) at appropriate values of E,,. Triply distilled H,O and D,O (MSDISOTOPES, > 99.8% purity), each degassed by repeated freeze-thaw cycles, were used for these experiments. The vapour was admitted via a Granville-Phillips leak valve to give an ambient pressure of 5 x lo-’ Torr. Lengthy equilibration in the case of D,O (no less than 24 hours) prior to data acquisition allowed near complete hydrogen-deuterium exchange on the chamber walls. Standard calibration runs for the 3p orbital of argon were performed to establish both the energy and momentum resolutions which were = 1.6 eV fwhm and = 0.15 a;’ fwhm, respectively. Under these conditions it was found, in accordance with earlier work [15], that the Ar 3p momentum distribution was quite well described within the PWIA and THFA using the Hartree-Fock limit wavefunction of Clementi and Roetti [45]. The measured momentum profiles (XMPs) for the four valence orbitals of H,O and also the three outer valence orbitals of D,O were put on the same (relative) absolute intensity scale (i.e. all relative normalizations preserved) by normalization on the peak areas in angular selected binding energy spectra [10,46]. This normalization involved integration over the full Franck-Condon width for production of each of the (lb,)-‘, (3aJ’, (lb,)-’ and (2aJ’ electronic states of H20+ yielding a relative ratio (at + = 8”) of 1.0 : 1.1 : 1.0 : 2.4. In this procedure it was necessary to take into account the full distribution of satellite structure * (25-45 eV) of the (2a,)-’ state [lo]. With this procedure a very stringent quantitative comparison (to better than 5%) with calculations is possible since experiment and the-

ory are normalized to each other at only a single point on one of the four measured momentum profiles. All other experimental and all other calculated points are therefore open to quantitative scrutiny.

4. Results and discussion 4.1. General considerations The new measurements of the momentum profile for the lb, orbital of H,O have been added to the results obtained for this orbital in the earlier reported study [lo]. This results in a considerable improvement in the quality of the data for this particular orbital which showed [lo] the greatest discrepancy between theory and experiment with the near Hartree-Fock level wavefunctions used in the calculations in our earlier study [lo]. This data for the lb, orbital of H,O and the existing data [lo] for the 3a,, lb, and 2a, XMPs for H,O (open squares, figs. l-4) have each been placed on the same relative intensity scales using the procedure outlined in section 3. Comparisons of the XMPs with calculated MDs and OVDs were made in a quantitative manner by a single point normalization to the MD calculated for the lb, orbital using the 109-GTO wavefunction which has the most accurate calculated properties for water at the Hartree-Fock level (see table 1). All other experimental and calculated data points (including the 109-G(CI) OVD results assuming unit pole strength, see section 4.4) for all four orbitals have

* In the binding energy spectra of many atoms and molecules, breakdown of the simple molecular orbital picture of ionization in the inner valence region has been both observed (see for example refs. [10,22]) and predicted [47]. This phenomenon has been attributed to many-body effects in the ionization process which can be described for example by final ion state CI. In the THFA the overlap between the ion and the neutral target is a linear combination of the occupied Hartree-Fock orbitals of the neutral molecule ground state. In many cases a single term in this linear combination dominates and the final state XMP can then be ascribed to a single initial state Hartree-Fock orbital. EMS cross sections have therefore been used to probe the origins of (energyresolved) many-body states [10,15,22].

A.O. Bawagan

ef al. / EMS of the valence orbiials of H,O and D,O

:

Oo.0 0.3

0.6

0.9 Momentum

1.5 1.2 (au.)

1.6

2.1

I 2.4

0.0

11

0.3

11

0.6

11 0.9 Momentum

1’ 1.2

11 1.5

I

0.0

0.3

0.6

0.9 Momentum

Fig. 1. Detailed comparison of the experimental momentum profiles (XMPs) of the lb, orbital of D,O (solid circles) and H,O (open squares) with several spherically averaged momentum distributions calculated using the THFA. Distributions calculated from the (1) SB [42/2], (2) NM [531/42], (3) NM (1062/42), (4) DF 84GTO, (5) 99-GTO, (6) 109-GTO, (6~) 109-G(CI) and (7~) the 140-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-CT0 calculation on the lb, XMP (see text for details). The sitting binding energy for the measurements is 12.2 eV. The instrumental momentum resolution (0.15 (I0 ’ fwhm) has been folded into the calculations.

x

x

I I 1.8

I 1 2.1

II 2.4

1.8

1.5 1.2 (a.lJ.)

‘Cd

2.1

2.4

Fig. 3. Detailed comparison of the experimental momentum profiles (XMPs) of the lb, orbital of D,O (solid circles) and H,O (open squares) with several spherically averaged momentum distributions calculated using the THFA. Distributions calculated from the (1) SB [42/2], (2) NM [531/42], (3) NM (1062/42), (4) DF 84-GTO, (5) 99-GTO, (6) 109-CT0 and the (6~) 109-G(U) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-CT0 calculation on the lb* XMP (see text for details). The sitting binding energy for the measurements is 18.6 eV. The instrumental momentum resolution (0.15 0;’ fwhm) has been folded into the calculations.

-_

0.1

1 2

SE NM

:

;w6u42)

[42/21 [531/421

6 6c

109-GTO 109-G(CI)

i

!.4

(au.)

Fig. 2. Detailed comparison of the experimental momentum profiles (XMPs) of the 3a, orbital of D,O (solid circles) and H,O (open squares) with several spherically averaged momentum distributions calculated using the THFA. Distributions calculated from the (1) SB [42/2], (2) NM [531/42], (3) NM (1062/42), (4) DF 84-GTO, (5) 99-GTO, (6) 109-CT0 and the (6~) 109-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-CT0 calculation on the lb, XMP (see text for details). The sitting binding energy for the measurements is 15.0 eV. The instrumental momentum resolution (0.15 no’ fwhm) has been folded into the calculations.

Fig. 4. Detailed comparison of the experimental momentum profiles (XMPs) of the Za, orbital of Ha0 (open squares) with several spherically averaged momentum distributions calculated using the THFA. Distributions calculated from the (1) SB [42/2], (2) NM [531/42], (3) NM (1062/42), (4) DF 84-GTO, (6) 109-CT0 and (6~) lOPG(C1) are placed on a common intensity scale by using a single point normalization of the 109-CT0 calculation on the lbs XMP (see text for details). The sitting binding energy for the measurements is 32.2 eV. The instrumental momentum resolution (0.15 a;’ fwhm) has been folded into the calculations.

28

A.O. Bawagan ei ul. / EMS of the valence orbit&

their absolute values relative to this single point normalization. The measured XMPs for H,O and D,O are shown on the same intensity scale in figs. l-4 in comparison with a wide range of already reported [lo] as well as new calculations. Some of the presently reported calculations involve the new, more sophisticated Hartree-Fock wavefunctions which are as yet unpublished (see section 2.2 above). These wavefunctions are used in the target Hartree-Fock approximation (THFA) to investigate basis set dependence and the convergence of calculated properties in addition to the normal variational treatment minimizing the total energy. Further calculations model the effects of correlation (with up to 88% of the ground state correlation energy recovered) by evaluating the full overlap amplitude (see eq. (1)) using CI wavefunctions for both the initial state target neutral molecule and the final ion state. The various results at the THFA level are discussed in the following sections with reference to figs. l-4. 4.2. Vibrational effects It can be seen from figs. l-4 that, within statistics, there is no obvious difference between the XMPs of H,O and D,O. There appear to be some possible slight differences in the high-momentum region (p 2 1.5 a&‘) of the lb, momentum profile where the results for D,O are slightly lower than those for H,O. However these differences are at best marginal considering the error bars involved. In earlier EMS work [38] no differences were observed between the XMPs of H, and D, which are the only other isotopically substituted molecules to be studied thus far by EMS. The present results for H,O and D,O indicate that vibrational effects have no significant effect on orbital XMPs at least in the case of water at the current level of experimental accuracy. The essentially identical XMPs observed for Hz0 and D,O now extend the earlier conclusions of Dey et al. [38] concerning the isotopic diatomic molecules, H, and D,, to the more complex polyatomic water system which has both symmetric and antisymmetric modes of vibration. It had been thought [38] that discrepancies between measured XMPs and the

of H,O and D,O

calculated MDs might result either from the failure of the Born-Oppenheimer approximation or from failure to consider accurately the vibrational integral in eq. (1). Several theoretical studies have investigated vibrational effects on calculated molecular properties. For example, a study of vibrational corrections to the Compton profile of H,O [48] suggested that the vibrational effect on MDs might be expected to be most pronounced, though small (less than l%), in the low-momentum region. The Compton profile was calculated [48] using the 39-ST0 wavefunction [34] for H,O (see later discussion) and vibrationally averaged over the SDQ (singly, doubly and quadruply excited CI) potential surface of Rosenberg et al. [49]. The present EMS results are also consistent with a series of studies [50,51] on vibrational corrections to the one-electron properties of H,O. It was noted [50,51] that these corrections are small and in the case of the dipole moment, the vibrational correction ranges from 0.3% in the ground state to 5% in the (0, 3, 0) excited vibrational state. A recent study by Breitenstein et al. [52] of vibrational corrections to electron impact differential cross sections also showed that such effects are small. Intuitively it would seem that even if vibrational effects on MDs are significant, the effect would be least on the (non-bonding) lb, orbital since it is effectively a lone pair mainly lying perpendicular to the molecular plane and thus it would be largely unaffected by the nuclear motion. The absence of momentum profile differences between Hz0 and D,O even for the bonding 3a, and lb, orbitals suggests that the presently observed discrepancies [lo] between calculations and experiment, particularly for the lb, orbital but also to a lesser extent for the 3a, orbital, are not due predominantly to vibrational effects. 4.3. Basis set effects Our recently reported study of the MDs of Hz0 in ref. [lo], as well as several earlier works, clearly indicated significant discrepancies between calculation using existing Hartree-Fock wavefunctions from the literature and experimental

A.O. Bawagan et al. / EMS

of the valence orbitals of H,O and D,O

MDs for the two outermost orbitals (lb, and 3a,). The greatest differences were observed for the (least tightly bound) essentially non-bonding lb, orbital. In contrast it was found that agreement for the third and fourth orbitals (lb, and 2a,) was very good at least for the relative shapes. In the earlier work [7], the wavefunctions used included the minimum basis set of gaussian functions of Snyder and Basch (SB [42/2]) [41] and the near Hartree-Fock (i.e. extended contracted gaussian) wavefunction of Neumann and Moskowitz (NM [531/21]) [43]. The details as well as properties of these and all other wavefunctions used in the present work are shown in table 1. The Slater type (STO) wavefunction of Aung, Pitzer and Chan (APC (331/21)) [42] which gives a total energy of -76.00468 au and a dipole moment of 2.035 D was also compared in the earlier work [lo] and gave the best agreement of the three wavefunctions (SB, NM and APC) with the experimental momentum profiles. These results of the earlier study [lo] showed that for H,O the lowest energy (i.e. best from a variational standpoint) wavefunction (contracted NM [531/21]) does not give the best calculated MD. In fact the variationally inferior APC wavefunction gave the better overall fit (but one that was still rather inadequate for the lb, and 3a, orbitals) to the measured XMPs. It is interesting in this regard to note that the APC wavefunction also gave a dipole moment (2.035 D) slightly nearer to the Hartree-Fock limit (1.98 k 0.01 D) than those given by either the contracted NM [531/21] (2.092 D) or SB (2.681 D) wavefunctions. From this rather limited comparison of theory and experiment it is clear that the calculated results for total energy, dipole moment and momentum distribution are far from converged (see tables 1 and 2 below). It is also clear that the degree of convergence is different for different calculated properties for each of the wavefunctions. This is, of course, a straightforward manifestation of the fact that a wavefunction is a model (in all cases for every molecule and for all neutral atoms except for atomic hydrogen). Such a model will only be as reliable as the approximations used in its building and it will only be adequate for calculating those properties for which these ap-

29

proximations and testing constraints (usually only the variational constraint of minimized energy) are sufficiently valid. Since the energy minimization stresses the small r region of wavefunctions it is not surprising that use of variationally determined wavefunctions often leads to poor results for properties, such as dipole moment and MDs, which depend sensitively upon the longer range (r) charge distribution. With the above considerations in mind, and considering the large discrepancy between measured XMPs and calculated MDs for the outer valence orbitals of water, it is of interest to investigate the effects of increased basis set flexibility including the addition of higher-order polarization functions. In this it is of key importance to ensure that convergence has been reached not only with regard to the HF limit of energy but also for those properties including the dipole moment and MD which are influenced by the large r (i.e. low momentum) part of the wavefunction. When this has been achieved, the remaining discrepancies (if any) between theory and experiment can be investigated in terms of the significance of the neglect of electron correlation and relaxation effects implicit in the Hartree-Fock model. At the time of the first EMS experiments for H,O the wavefunctions used were in the range of quality of the SB, NM and APC functions referred to above (see also table 1). Our earlier EMS results, at much improved momentum resolution, were evaluated [lo] using these same wavefunctions in order to facilitate direct comparison with the original studies [7,8]. Other more sophisticated literature wavefunctions are now compared with experiment. These include the 39-ST0 wavefunction of Rosenberg and Shavitt [34] and the 84-GTO wavefunction of Davidson and Feller [33] which are, we believe, the best single-determinant SCF wavefunctions currently published for H,O. These two wavefunctions have total energies within 0.003 and 0.001 au, respectively of the estimated HF limit at the experimental geometry and give calculated dipole moments of 1.995 and 2.021 D which are close to the estimated HF limit of 1.98 f 0.01 D (experimental value is 1.8546 D). It is however of interest to note that the variationally superior 84-GTO wavefunction gives a less good

30

A.O. Bawagan et al. / EMS of the valence orhitals of Hz0 and D,O

value (2.021 D) for the dipole moment than the 39-ST0 (1.9951 D). This may reflect the superiority of STOs at large distances. Consideration of these two relatively high quality wavefunctions [33,34] alone indicates that convergence of both energy and dipole moment to the HF limit values has not yet been reached. Calculations in fact show that the 84-GTO [33], 39-ST0 [34] and APC [42] wavefunctions all give quite similar results for the calculated MDs exhibiting in each case a similar (considerable) discrepancy with experiment for the lb, and 3a, orbitals of H,O (compare results in ref. [lo] and figs. 1 and 2 of the present paper). Therefore new and further improved wavefunctions, namely 99-GTO and 109-GTO, have been generated in the course of the present work in an attempt to model more adequately the variationally insensitive but chemically important large-r (low-p) portion of the electron distribution. Details of these new wavefunctions are given in section 2 and pertinent properties are shown in table 1. To give an idea of the basis extension, it should be noted that the 109-GTO wavefunction uses a [14slOp3dlf/6s3pld] basis set whereas a minimal basis set for water would use only a [2slp/ls] basis set. It can be seen that the energy (109-GTO) is converged (at least to within 0.0005 au of the HF limit) but that the dipole moment (2.006 D) is still farther from the HF limit than those given by the variationally inferior NM (1062/42) [43] and 39-ST0 [34] wavefunctions. Even larger gaussian basis sets than those reported here have given a Hartree-Fock dipole moment of 1.9803 D with an energy of - 76.0672 hartree. This value of the dipole moment is believed to be converged to kO.01 D. Calculations of the momentum distributions for all four valence orbitals (including spherical averaging and incorporation of the experimental momentum resolution) using these various existing and new wavefunctions are shown in figs. l-4 in comparison with the experimental results for H,O and D,O. The calculations were carried out in the PWIA and (except for the 109-G(C1) and 140G(CI), see below) THFA treatments. All calculations (including the lOPG(C1) assuming unit pole strength for each orbital) are on a common inten-

sity scale established by single point normalization of the 109-GTO calculations to experiment on the lb, orbital (see fig. 3). Thus all experimental points and calculations are on the same relative absolute intensity scale for all four orbitals and this affords a very stringent quantitative comparison of theory and experiment. Several observations can be made in reference to the earlier comparisons [lo] of experiment and theory for the valence orbitals of H,O. It should be noted that curves 1 (SB) and 2 (NM[531/21]) were shown [lo] with individual height normalizations for each orbital to the experiment. Comparison of curves 2 and 3 shows the serious effect of the contraction of the same (NM) wavefunction on the calculated MDs particularly in the case of the lb, and 3a, orbitals. The uncontracted set, NM(1062/42), shows (see curve 3, figs. 1 and 2) an improved momentum distribution and a much better dipole moment than does the contracted set, NM[531/21] (see curve 2, figs. 1 and 2) although the energy is only marginally affected by the contraction (see table 1). This illustrates the extreme care necessary in choosing the proper contraction scheme if properties such as the MD or dipole moment are required. Curve 4 (figs. l-4) shows in each case the MDs calculated from the 84-GTO wavefunction and these are found to give rather similar results to the APC wavefunction as used earlier [lo]. Considering first the lb, orbital (fig. 1) it can be seen that the increase in the maximum cross section and decrease in p,, as well as the increasingly improved modeling of the low-momentum region in going from SB (curve 1) to NM(1062/42) (curve 3) to 84-GTO (curve 4) are carried even further in going to the 99-GTO (curve 5) wavefunction. Similar improvements with change in basis set also occur for the 3a, (fig. 2) and 2a, (fig. 4) orbitals. This reflects the further improvement (see table 1) in both calculated total energy and dipole moment for the 99-GTO and 109-GTO wavefunctions relative to the 84-GTO wavefunction. The progression of the theoretical MDs towards the experimental momentum profile with expansion in basis set is clearly illustrated. In contrast, the calculated MDs for the lb, orbital (curve 3, fig. 3) converge at the NM(1062/42)

A.O. Bawagan et al. / EMS of the valence orbit&

level. However, it is of importance to note that no further change in the calculated MDs for the two outermost orbitals occurs in going from 99-GTO to 109-GTO (see curves 5 and 6 which are identical in figs. l-3). On the other hand, in the case of the 2a, orbital the calculated MD is already converged at the 84GTO level (see fig. 4). Even larger basis sets (114-GTO, 119-GTO and 140-GTO) gave no noticeable change so these MD curves would seem to be at the Hartree-Fock limit. These results indicate the importance of having s-p saturated basis sets as well as higher-order polarization functions (i.e. d- and f-functions on the oxygen) when predicting properties such as electron momentum distributions and the dipole moment. The importance of very diffuse functions (much more than expected) in the basis set is shown by the corresponding improvement in the calculated MDs. As more diffuse functions are added, the MD shifts and also gives appreciably more intensity at lower momentum. The balanced addition of extra diffuse functions in H,O evidently provides an improved description of the large-r (low-p) part of the total wavefunction. In terms of the inverse weighting property of the Fourier transform this should contribute to a better description of the low-momentum components of the calculated MD. Summarising the above considerations it can be concluded that gaussian basis set saturation has been effectively reached at the 99-GTO SCF level. However, while it is clear that considerable improvement over earlier calculations has been gained for the lb,, 3a, and 2a, orbitals with this “best” HF level treatment it can be seen that a considerable discrepancy with experiment still occurs, especially in the low-momentum region. In particular, significant additional low-momentum components are observed experimentally in the case of the lb, orbital and also the observed p,, (0.60 f 0.02 a;‘) is appreciably lower than even that (0.65 a; ‘) predicted by the best (99-GTO and 109-GTO) HF level wavefunctions. Smaller discrepancies exist between calculations and experiment for the 3a, and 2a, orbitals at the HF level. Properties of the various calculated MDs and XMPs, including p,,,, , are given in table 2. The properties shown in this table summarize the con-

of H,O and D20

31

vergence of the calculated molecular orbitals of water in momentum space. To characterize the MDs two properties have been evaluated, namely the leading slope at half maximum (LSHM) and the momentum at which the MD maximizes (P,,) *. The pmax and the LSHM of the MDs and XMPs can be considered as momentum space analogs of one-electron properties in position space in the sense that they can be used as “diagnostics” of wavefunction quality. The pm, and LSHM values for the MDs were obtained using a standard cubic spline fitting routine [53]. The statistical deviation (i.e. the individual error bars of the points in the distribution) were also considered in obtaining the best fits for the XMPs. It can be seen that the experimental p-space properties for H,O and D,O are the same within experimental error. Consider first the trend of p,_ and LSHM at the SCF level. From table 2 it is evident that the for the 3a, and lb, orbitals are quite conP vz;ed at the 99-GTO level and NM(1062/42) level (58-GTO), respectively with the lb, orbital converging towards a pm, higher (0.72 ai’) than the 3a, orbital (0.68 a;‘). In contrast, the p,_ of the lb, orbital converges at the 99-GTO level with a Pmax of 0.65 ai* compared to the experimental value of 0.60 + 0.02 a;‘. Despite the estimated uncertainty it is clear from the trend of the data points (fig. 1) that the p,,,, from experiment is significantly lower than that predicted by any calculation at the THFA level. Since basis set saturation has been established the remaining discrepancy between theory and experiment may be associated with one or more of the following effects: (1) Deficiencies in the PWIA treatment. (2) Further uncertainties in the experimental momentum resolution beyond Ap = 0.15 a;’ fwhm already incorporated in the calculations. (3) Failure of the THFA in H,O and therefore the need to consider the fact that correlation and It should be pointed out that p,_ is not the most probable momentum which is characterized by the p which maximizes p*.(MD). Note that the pmax and the LSHM do not completely characterize the MD but they can be used as guides in comparing the different calculations with experiment.

A.O. Buwagan et al. / EMS

32 Table 2 Characteristics Basis set

‘) of calculated

orbital

momentum

of the vulence orbit& of Hz0 and D,O

distributions

and experimental

momentum

lb,

3al

lb,

profiles

PlL%X

LSHM

PlKiX

LSHM

(0.0990) (0.0989) (0.1121) (0.1195) (0.1204) (0.1262) (0.1264)

0.75 0.76 0.71 0.69 0.69 0.68 0.68

(0.1104) (0.1078) (0.1186) (0.1240) (0.1270) (0.1280) (0.1280)

0.74 0.74 0.72 0.72 0.72 0.72 0.72

(0.1212) (0.1192) (0.1216) (0.1204) (0.1210) (0.1208) (0.1209)

0.67 0.63

(0.1316) (0.1398)

0.68 0.66

(0.1365) (0.1370)

0.72 0.72

(0.1222) (0.1216)

0.60 0.60

(0.115) (0.115)

0.69 0.69

(0.110) (0.110)

0.72 0.72

(0.110) (0.110)

PlMX

LSHM

SB [42/2] NM [531/21] NM (1062/42) RS 39-ST0 DF 84.GTO 99.GTO 109-GTO

0.16 0.76 0.72 0.67 0.68 0.65 0.65

84-G(C1) 109-G(C1) Hz0 expt. b, D,O expt. b,

a) Peak maxima and LSHM (in parentheses) are both quoted in atomic units. ‘) Estimated experimental uncertainty in pmax and LSHM are +0.02 ai1 and kO.005,

relaxation effects may be significantly influencing the valence momentum profiles. This would amount to failure of the Hartree-Fock model description in these cases. Deficiencies in the PWIA treatment (1) are unlikely for the reasons already discussed in the introduction (i.e. good agreement at the HF level for a number of other atoms and molecules). Similarly unknown momentum resolution effects (2) can be discounted since a further enlargement (unphysical with respect to the experimental geometry) of the Ap (0.15 n;’ fwhm) already used would in any case only significantly affect the very low momentum part of the curve. Any such increase in Ap would not change the position of p,, or the majority of the large “mismatch” down the leading edge (i.e. low-p region) of the momentum distribution (see fig. 1). Therefore the most likely source of the remaining discrepancy between HF theory and experiment is item (3), namely the neglect of electron correlation implicit in the Hartree-Fock model used in the THFA treatment. This possibility is investigated in detail in section 4.4 following. Before proceeding to a consideration of correlation effects the following further observations are made. First, while the observed discrepancies between calculated MDs and measured XMPs for H,O are largest for the lb, orbital similar but

respectively.

somewhat smaller discrepancies are also found for the 3a, and 2a, orbitals. On the other hand, the lb, orbital is apparently well represented already at the APC and NM(1062/42) levels. Second, it is instructive to compare MD results using the best gaussian basis set (i.e. 109-GTO or 99-GTO) and best Slater [34] basis sets (39STO) available to date. This “best” GTO/STO comparison is shown together with the experimental measurements in fig. 5 for the valence orbitals of H,O and D,O. The overall good agreement between the two calculations confirms the generally held view that 2 to 3 GTOs are needed for each STO. It can be seen that while the calculated lb, orbital is identical for use of both ST0 and GTO (fig. 5) and in good agreement with experiment, the 109-GTO calculation is marginally better for the 3a, orbital and a slight improvement for the lb, orbital. It can also be seen from tables 1 and 2 that although the 109-GTO wavefunction gives superior values for calculated total energy, p,, and (J-“) the 39-ST0 wavefunction gives a better value for the dipole moment. 4.4. Correlation and relaxation

effects

In view of the failure of even highly saturated basis sets to satisfactorily predict the observed momentum distributions at the target Hartree-

33

A.O. Bawagan ei al. / EMS of the valence orhitals of H>O and D20

3

“Cj

SN

4 zq 5 OO.0 E Y

9 0.5

1.0

1.5

2.0

2.5

OO.0

0.5 I

I

1.0 ,

1

1.5 I

1

2.0

2.5

Momentum (a.~.) Fig. 5. Comparison of calculated valence orbital MDs of H,O using the best gaussian (109-GTO, solid line) and best Slater (39~STO, broken line) [34] basis sets. The THFA calculations are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the lb, XMP (see text for details). The instrumental momentum resolution (0.15 ai’ fwhm) has been folded into the calculations.

Fock level (except for the lb, orbital, at least on the basis of the present normalization) a theoretical investigation beyond the Hartree-Fock model has been made using CI wavefunctions. The HF limit of total energy for the ground state of H,O is estimated to be -76.0675 au while the estimated non-vibrating total energy is non-relativistic, - 76.4376 au [34]. The difference (- 0.370 au) is the extra energy due to electron correlation neglected in the Hartree-Fock single configuration SCF model. Correlation effects can be treated by

configuration interaction (CI) description for the target molecule. For the final ion species, configuration interaction using the target molecular orbitals can be used to describe both electron correlation and relaxation effects. These correlated wavefunctions can then be used to evaluate the full ion-neutral overlap (see eq. (1)). Any difference between such a full overlap and the corresponding THFA calculations indicates the importance of relaxation and correlation effects. Previous comparisons of XMPs and theoretical

34

A.O. Bawagan et al. / EMS of the valence orbitals of Hz0 and D,O

MDs for Hz0 were all done using the target Hartree-Fock approximation (THFA) except for the generalized overlap amplitude (GOA) calculation of Williams et al. [27]. These GOA calculations [27], which utilized a rather limited basis set, showed a small improvement over the THFA calculation but major discrepancies between experiment and theory remained. The GOA method attempts calculation of the ion-neutral overlap via many-body Green function techniques. In the present work we have performed calculations of the ion-neutral overlap amplitude using CI wavefunctions based on the two highly extended basis sets, namely the 84-GTO [33] and 109-GTO basis sets. For the CI treatment using the 84-GTO basis (labelled 84-G(C1)) the neutral molecule (‘A,) wavefunction is expanded into 5119 symmetry adapted configurations (SACS) calculated at the experimental neutral molecule equilibrium goemetry (EC, = -76.3210 au which includes 69% of the total ground state correlation energy). For the CI treatment using the 109~GTO basis (labelled 109-G(CI)), the neutral molecule is expanded into 11011 SACS (EC, = -76.3761 au which includes 83% of the ground state correlation energy). Details of the 109-G(C1) final ion state wavefunctions are to be found in table 3. The spherically averaged square of the Fouriertransformed ion-neutral overlap amplitudes using both sets of CI wavefunctions have been calculated and the momentum space properties are compared in table 2. The full ion-neutral overlap calculations (i.e. 84-G(C1) and 109-G(C1)) are compared to the corresponding THFA results in

figs. 6-8 (see also figs. l-4) along with the measured H,O and D,O XMPs for the lb,, 3a,, lb, and 2a, valence orbitals. Normalization is the same as in the case of figs. l-4 (i.e. at a single point for experiment and the 109-GTO (THFA) calculation on the lb, momentum profile). Comparison of the 0 o and 8 o binding energy (at a total energy of about 1200 eV) spectra [9,10] as well as the momentum profiles measured at 32.2 and 35.6 eV [lo] have confirmed the assignment that the extensive many-body structure found in the region 25-45 eV is predominantly due to the (2aJ’ hole state. CI calculations however suggest the presence of minute poles in this region due to small contributions from the ionization of the outer valence orbitals. This predicted spreading of the minor poles over the wider energy spectrum (with much of the extra intensity at energies even higher than 45 eV) results in pole strengths for the main lines of each of the three outer valence orbitals which are slightly less than one (i.e. 0.87, 0.88 and 0.89 - see table 4). These pole strengths for the outer valence orbitals are converged to & 1%. Even more extensive breakdown of the singleparticle picture in the case of ionization of the 2a, inner valence orbital is well known both theoretically [47] and experimentally [9,10]. Table 4 shows the pole strengths and energies for each final ion state together with the CI coefficients for 2a,/3a, mixing as calculated using the lOPG(C1) correlated wavefunctions. These calculated pole strengths are convoluted with the estimated experimental width (2.77 eV) of the main (2a,)-’

Table 3 CI calculations (au) of the ground and final ion states of H,O using the 109-GTO basis set AE(CI) b,

E(exp.) ‘)

E(HF)

- 76.0671

0.309

- 76.4376

- 75.5569 - 75.4822 - 75.3492

0.364 0.351 0.336

- 75.9746 - 75.8916 - 75.7576

-

State

E(CI)

E(SCF)

H,O X’A,

- 16.31614

H;O+ (lb;)-’ HaO+ (3a,)-’ H,O+ (lb,)-’

- 75.92084 - 15.83376 - 75.68563

‘) b, ‘) d, ‘) e

a)

For the ionic states the energy refers to a Koopmans energy. AE(C1) = E(CI) - E(SCF). For the ion states, E(exp.) = - (76.4376 - IP) au. Includes relaxation for ionic states. E(cmr) = E(exp.) - E(HF) for neutral, E(corr.) = E(exp.) - E(Koopmans) SE(cm) = AE(CI)/E(corr.)xlOO.

=)

76.0615 75.5600 75.4841 75.3515

for ion.

E(corr.) 0.370 0.415 0.414 0.406

d*e)

%E(corr.) d,n 83 88 85 83

A.O. Bawagan et al. / EMS

I 4 4c 6

84-GTO 84-qCI)

109-GTO

1 0.0

0.3

0.6

0.9 1.2 Momentum (au.)

1.5

1.8

2.1

35

of the valence orbitals of H20 and D,O

J

2.4

Fig. 6. Correlation effects in the calculated momentum distributions of the lb, orbital of water. The more accurate CI overlap calculations (solid line), (4c) 84-G(CI), (6~) 109-G(CI) and (7~) 140-G(U) and the THFA calculations (broken line), (4) 84-CT0 and (6) 109-CT0 are all placed on the same relative absolute scale by using a single point normalization of the 109-CT0 calculation on the lb, XMP (see text for details). Further comparisons are made with the measured H,O (open squares) and D,O (solid circles) experimental momentum profiles. The instrumental momentum resolution (0.15 a;i fwhm) has been folded into the calculations.

peak (see figs. 1 and 2 of ref. [lo]) and shown in fig. 9 below in comparison with the relevant part of> the earlier reported binding energy spectrum [lo] in the region 23-45 eV. The agreement between the calculation and experiment is generally quite reasonable. However, the calculated results are at best semi-quantitative with respect to the distribution of intensity particularly in the 31-36 eV region. Note that the calculated binding energy profile is slightly shifted (= 1 eV) to the higher energy side and that the position of the shoulder at = 35 eV is still inadequately predicted by the 109-G(C1) calculation. It should also be noted that the spectrum reported by Cambi et al. [9] extends out to 50 eV and shows weak intensity comparable to that predicted at = 47 eV. It is probable that, even at this level, the calculation is still basis set dependent with respect to the exact (2a,)-’ pole strength distribution (see also, for example, pole strength Ocalculations reported by Cambi et al. [9] and Agren and Siegbahn 1541 which are shown in comparison with experiment in fig. 2 of ref. [lo]). More definite conclusions

ll

2.4

Momentum

(au)

Fig. 7. Correlation effects in the calculated momentum distributions of the 3ai orbital of water. The more accurate CI overlap calculations (solid line), (4c) 84-G(C1) and (6~) 109G(CI), and the THFA calculations (broken line), (4) 84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the lb2 XMP (see text for details). Further comparisons are made with the measured Hz0 (open squares) and D,O (solid circles) experimental momentum profiles. The instrumental momentum resolution (0.15 a&,’ fwhm) has been folded into the calculations.

__

Oo.0

I 0.3

0.6

0.9 Momentum

1.2

1.5

1.8

2.1

2.4

(au)

Fig. 8. Correlation effects in the calculated momentum distributions of the lb, orbital of water. The more accurate CI overlap calculations (solid line), (4c) 84-G(CI) and (6~) 109G(CI), and the THFA calculations (broken line), (4) 84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the lb, XMP (see text for details). Further comparisons are made with the measured H,O (open squares) and D,O (solid circles) experimental momentum profiles. The instrumental momentum resolution (0.15 a;’ fwhm) has been folded into the calculations.

36

ofthe valence orbitals of HI0 and D20

A.O. Bawagan et al. / EMS

would require a convergence study of the energies and pole strengths of the many-body structure of the (2a,)-’ hole state. With these considerations in mind, we have chosen to present the calculated outer ualence OVDs (figs. 1-3, 6-8) as normalized distributions (i.e. renormalizing the pole strength of each of the outer valence poles to unity). The inner valence 2a, OVD in fig. 4 was obtained by calculating a pole-strength weighted sum of the (slightly different) OVDs of all significant poles (i.e. those with intensity greater than 1%) found in the region above 25 eV and then divided by the summed pole

Table 4

~1 calculations (109~G(C1)) of the pole strengths and energies (ev) in the binding energy spectrum of Hz0

State

Energy

Pole strength

C2(2a,)

C2(3a,)

(lb,)-’ (3a,)-’

0.869 0.882 0.888

0.000

0.998

(lb,)-’

12.39 14.76 18.79

(2a,)-’

29.0

0.0357 *

0.974

0.021

33.1 33.6 34.2 35.8 37.8 39.5 41.0 44.0 44.9 45.0 45.8 46.3 47.1

0.4394 * 0.2232* 0.0225 * 0.0218’ 0.0216 * 0.0730 * 0.0240 0.0039 * 0.0002 0.0009 0.0051* 0.0043 0.0227 *

0.992 0.999 0.659 0.865 0.863 0.980 0.475 0.708 0.336 0.033 0.914 0.393 0.827

0.077 0.000 0.325 0.133 0.135 0.015 0.487 0.260 0.525 0.482 0.058 0.504 0.147

strength

in this region

in order

to renormalize

the

resultant OVD to unity. It is of interest to note that this summed 2a, pole strength has essentially the same value

as that

the three

valence

tion

procedure

outer tions

Z(*) = 0.869

outer

for the main

any contributions

from

valence poles have in any case indicate

been that

109-G(

Cal cn.

Cl )

pole of each of

In this normaliza-

orbitals.

neglected. such pole

any minor Calculastrengths

1

I

I

I

I,

I

I

I

I

I,

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

23.0

25.0

27.0

29.0

31.0

33.0 35.0 37.0 Binding Energy (N)

39.0

41.0

43.0

45.0

47.0

49.0

23.0

25.0

27.0

29.0

31.0

33.0 35.0 37.0 Binding Enargy (eV)

39.0

41.0

43.0

45.0

47.0

49.0

Fig. 9. Binding energy spectrum of water in the (2a,)-’ inner valence region. The experimental data (a) at $I= 0” (E, = 1200+ BE eV) are taken from ref. [lo]. The theoretical binding energy profile (b) was determined in the present work from pole strengths calculated using the 109-G(C1) wavefunctions (shown as solid vertical lines at the appropriate energies). The calculated poles have been convoluted with an experimental width of 2.77 eV fwhm estimated from spectrum (a). Further details are given in table 4.

A.O. Bawagan

et al. / EMS

of the valence orbitals of H,O and D,O

are less than 3% (in fact only one pole is >, l%, i.e. that at 41 eV, see table 4). On the basis of the above normalization procedure for the CI calculations, it can be seen (figs. 6 and 7, table 2) that use of the CI wavefunctions for the lb, and 3a, outer valence orbitals in the overlap treatment results in a further significant progression of these “p-type” calculated OVDs towards low momentum and higher cross section and thus towards better agreement with experiment. Such a shift towards low momentum with the inclusion of correlation has also been observed earlier for Ar (3~) by Mitroy et al. [21] but the effect was very small in that case. It is also significant to note that the good agreement already attained at the 84-GTO THFA level for the lb, orbital (fig. 8) is unchanged by use of the 84-G(C1) or 109-G(C1) overlap treatments. It can be seen that, with the normalization procedure described above, the THFA and CI treatments give the same absolute intensities over the entire momentum range for the lb, orbital. In the case of the 2a, inner valence orbital (fig. 4) inclusion of correlation (109-G(C1)) gives a slightly improved quantitative fit to the data. These findings indicate the important fact that correlation mainly influences the lb, and 3a, orbitals. This behavior is in sharp contrast to the situation for the lb, orbital where correlation and relaxation effects appear to have a negligible effect on the calculated overlap distribution. The interplay of basis set effects and the inclusion of correlation are well illustrated by the results for the lb, and 3a, orbitals shown in figs. 1, 2, 6 and 7. It can be seen that inclusion of correlation through the CI overlap treatment further improves agreement between the calculated distributions (MDs and OVDs) and experiment at both the 84-GTO level (compare curves 4 and 4c, figs. 6 and 7) and the 109-GTO level (compare curves 6 and 6c, figs. 6 and 7). In comparison with the 84-G(C1) wavefunction (p,, = 0.67 a,‘, dipole moment = 1.929 D) use of the lOPG(C1) wavefunction shows a further significant shift of the OVD to low momentum (pm, = 0.63 a;‘) together with a further increase in cross section and an improved value of the dipole moment (1.895 D). Despite the dramatically improved

31

agreement over the THFA treatments a small but significant discrepancy still exists for the lb, orbital even at the 109-G(C1) level since the experimental momentum profile is at lower momen= 0.60 f 0.02 a;‘) with a higher maxitum (P,, mum cross section. In the case of the 3a, orbital the 109-G(C1) overlap calculation gives a result quite close to experiment. With inclusion of configuration interaction, smaller p,, values are predicted together with more asymmetric OVD curves as characterized by the higher LSHM values (i.e. steeper leading slopes at half maximum) which, for example, for the lb, orbital are 0.1316 and 0.1398 for the 84-G(C1) and 109-G(C1) wavefunctions, respectively. The experimental LSHM for H,O (lb,) is 0.115 &-0.005 au. Calculations with even larger basis (llCG(CI), 119-G(C1) and 140-G(C1)) which recovered even more of the correlation energy of the neutral molecule and of the cation gave OVDs indistinguishable from the 109-G(C1) results for the lb, orbital. These considerations together with the fact (see table 2) that the 84-G(CI), 109-G(C1) and 140-G(C1) treatments recover 69, 83 and 88% respectively of the estimated correlation energy of H,O (-0.370 au), suggest that any further (very difficult) theoretical investigations of the effects of the remaining correlation energy on the small residual discrepancies are not likely to produce much change. In this regard it should be noted that the large shift (curves 4c and 6c, fig. 6) which results in going from 84-G(C1) to 109-G(C1) is mostly due to the improvement in the basis set. A smaller part of the shift can be ascribed to improvement in the CI description. It is also of interest to consider the various calculations for the lb, orbital as shown in figs. 1 and 6 together with the experimental results. In particular it is noticeable that all calculations are slightly higher than the measured cross sections in the 0.9-1.2 a,’ region of the XMP for the lb, orbital. The shift of the calculated OVDs towards low momentum using the full overlap treatment (i.e. at the CI level) reflects the “non-characteristic” contributions in addition to the single-particle orbital MD that would have been expected in the simple approximation of the more usual THFA. Consider the case of the (lb,))’ ion electronic state. The

A.O. Bawagan et al. / EMS

38

of the valence orbitals of H,O and D,O

overlap amplitude can be expressed in terms of the single-particle orbitals (of the ground Hartree-Fock configuration), (p’k,N-‘l~~)=c,~,+c*~,+...,

(3)

where the labels 1 and 2 refer to lb, and 2bi p-space molecular orbitals, respectively. The first term corresponds to the characteristic orbital (in the nomenclature of THFA) whereas the second term corresponds to the lowest-lying virtual orbital of the same symmetry. The latter can therefore be considered as a “non-characteristic” contribution. The correction comes mostly from the overlap integral of the lb: + 2b: double excitation of the neutral (which represents radial electron correlation) with the lb, --, 2b, single excitation of the ion (which describes orbital contraction). Because of the signs involved, this orbital contraction makes the molecule-ion overlap more diffuse than the characteristic orbital. To illustrate this point consider a two-orbital model: I\k,N)= Arllb:)

- A,1 2bf).

(4)

There will be no large term in (lb,2b,) because of Brillouin’s theorem [55]. The 2b: term represents electron correlation. A, and A, will be positive, independent of the relative phases of lb, and 2b,. For the ion, l!I’;-‘)

= B, ]lb,)

- B, ]2b,).

(5)

The second term represents relaxation (i.e. orbital contraction). The sign of B/B, will depend on the relative phases of lb, and 2bi. If both lb, and 2b, are chosen to be positive at large r, then B,/B, will be positive. Since lb, has no radial node and 2b, has one, they will have opposite signs at small r. Consequently, the second term will subtract from the first at large r and add at small r. 1?Py-‘) will be contracted relative to Ilb,). Now consider the molecule-ion overlap (\k,N-’ I et). Explicit evaluation gives (*:-‘I*,)

=A,B, =

Ilb,)

C, Ilb,)

+ AZB2 ]2b,) +

C,]2b,).

Notice that C, has the opposite sign to -B,

(6) so

that the molecule-ion overlap amplitude is more diffuse in position space than lib,), opposite to the contraction in I !Pr-‘). This behaviour and the converse momentum space contraction is clearly illustrated by considering the difference density ( PCI - PTHFA ) plot for the lb, orbital shown in fig. 10. Likewise the overlap density (the spherical average of which yields the OVD) can be written as pO = C:(lb,)*

+ C;(2b1)*

+ 2C,C2(lb1)(2b,). (7)

Since lB21 < IB,l, iA21 < IA,l, we find ]C,( < ] C, ] and C: is small compared to 2C,C2. Consequently the cross term in (lb,)(2b,) is the dominant correction in the molecule-ion overlap density. This momentum space contraction on going to the CI overlap treatment can also be clearly seen in the distributions shown in figs. 1 and 6. 4.5. Calculated properties

near the HF and CI limits

To summarize the various basis set, relaxation and correlation effects discussed above, the p,,, and LSHM of the theoretical MDs and OVDs of the lb, orbital are plotted as a function of the number of contracted GTOs in their respective basis sets in fig. 11. Attention is focused on the lb, orbital since it is most sensitive to correlation and basis set effects. As has been shown above, the 3a, and 2a, orbitals are also quite sensitive to these effects. The total energy which has been used traditionally as the principal diagnostic for wavefunction quality is shown as a function of the number of contracted GTOs in fig. lld. The estimated Hartree-Fock limit and experimental energy are represented by dashed lines. It is of great importance to note that the convergence of the calculated momentum space properties (p,, and LSHM) as a function of basis set (figs. lla and llb) is somewhat slower than that for the total energy. Whereas for most purposes one can consider the total energy to be converged essentially at the HF limit (see table 2) at the 84-GTO level the same is not true for the momentum space properties which converge at the 99-GTO level as shown for the lb, orbital of water (figs. lla and

39

A.O. Bawagan et al. / EMS of the valence orbitals of H,O and D20

MOMENTUM

r

DENSITY

DIFFERENCE

POSITION

DENSITY

DIFFERENCE

Fig. 10. Two-dimensional density difference ( pc, - pTHFA) plots in momentum space and position space for the lb, orbital of an oriented water molecule calculated using the 84GTO basis. Contours are at k 80, f 40, * 8, f 4, + 0.8, f 0.4, f 0.08 and f 0.04% of the maximum density difference. Positive difference is given by the solid lines, negative difference by the dashed lines. All dimensions are in atomic units.

llb). The convergence of the dipole moment (fig. llc) is less regular by comparison to the p,,, LSHM and total energy. These differences in convergence are not unexpected in view of the fact that the error in a calculated molecular property (e.g. the dipole moment or the momentum distribution) is first order with respect to the error in the wavefunction whereas the error in the total energy is second order [35]. The findings of the present work stress the need for proper use of the variational theorem, i.e. the constraint of energy minimization must always be accompanied by correct prediction of an adequate range of molecular properties. Since the lowmomentum regions of the orbital MDs include contributions from regions of r-space which make little contribution to the total energy, it is not unreasonable to expect that the XMPs may be inadequately described by methods which emphasize only the energetics of the system when choosing the basis set. While wavefunctions more extended than 109-GTO have been constructed [34,35] they cannot give more than 0.0005 hartree improvement to the RHF energy since the Hartree-Fock limit has already been reached within that accuracy (see table 2 and fig. lld). The present 109-GTO basis is also saturated with diffuse basis functions (at least at the HF level) so

further significant changes in the MD are not obtained at the THFA level. Finally it should be pointed out that comparisons of XMPs and calculated MDs and OVDs demonstrate the extreme sensitivity of the EMS technique towards certain details in the electronic wavefunction to which the energy is much less sensitive. As has been shown by EMS, agreement of theoretical orbital MDs with XMPs can be obtained for more tightly bound orbitals such as the lb, and 2a, (and probably also la,) orbitals already at the simple DZ level. These distributions could then tend to dominate the total electron momentum distribution and thus no serious discrepancy might be detected between theory and experiment for the total momentum distribution as observed for instance in the Compton profile [36,56]. The present work clearly demonstrates the orbital specificity of EMS which provides in more detailed information than methods such as Compton scattering [36,56] which measure the total momentum distribution. Improving the accuracy of calculated orbital MDs and OVDs would be of importance not only from the computational point of view but also with respect to the interpretation of current EMS experiments. In this regard, recent theoretical efforts towards the solution of the Hartree-Fock

A.O. Bawagan et al. / EMS of the valence orhitals of H,O and D,O

40

3 4

0

40

80

NMU062/42) OF S4-GTO

4c E4-G(U) 6c IDS-GICU

120

Number of Contracted GTOs Fig. 11. Convergence of calculated properties, (a) pm, and (b) leading slope at half maximum (LSHM), (c) dipole moment and (d) the total energy as a function of basis set complexity. LSHM and dipole moment are plotted on the The P,,~ left-hand side while the total SCF Hartree-Fock energy is plotted on the right-hand side. The estimated Hartree-Fock limit and total experimental energy are shown by the dashed lines in (d). Also shown in the lower half (note different energy scale) of (d) are the respective CI energies of the W-CT0 and 109-CT0 basis sets. For comparison, the experimentally observed pm,, LSHM and dipole moment are shown in the extreme right of (a)-(c), respectively. The broken vertical line in (a)-(c) separates the single configuration (Hartree-Fock) and many-configuration (CI) results. All quantities are quoted in atomic units except for the dipole moment which is quoted in debyes (D).

equations in momentum space [57], alternative theoretical approaches [58] other than the “minimal energy criterion” and also new ways to model molecular wavefunctions would be of interest. On the other hand, the design of momentum density optimized small basis sets, although computationally attractive, will lose physical meaning if the energy is non-optimal. 4.6. Summary

and conclusions

Considerable improvement has been obtained in the degree of agreement with experiment of

THFA calculations of MDs using wavefunctions at the Hartree-Fock limit in the case of the two outermost orbitals (lb, and 3a,). However, at the THFA level there still remains an appreciable discrepancy most notably in the case of the lb, orbital. Incorporation of correlation and relaxation effects by calculation of the ion-neutral overlap distribution using CI wavefunctions for the initial and final states results in generally very good agreement with the EMS data for all four valence orbitals. However small discrepancies still exist, especially for the lb, orbital. With both the highly extended basis sets as well as with the successively improved calculations of the ion-neutral overlap amplitude using CI wavefunctions, the shift of the theoretical MDs and OVDs has consistently been towards lower p,, and higher cross section, and thus toward better agreement with experiment. It is noteworthy that an improved theoretical description of the experimental momentum profiles occurs when there is an accompanying improvement in the prediction of both the total energy and the dipole moment as well as other properties. This has also been observed earlier in the case of NO [28]. The present work shows that it is important to perform the overlap calculations with “near-complete” basis sets - a method which is computationally very difficult. Investigation of whether the remaining small but finite difference between the measured XMP and calculated MDs and OVDs for the lb, of H,O is due to the still unaccounted for part of the electron correlation energy (88% of the correlation energy is accounted for in the 140-G(C1) calculation), or to some other factor such as a breakdown of the PWIA due to distortion of the incoming and outgoing electron waves by the polar target molecule will have to await further developments in quantum mechanical computation and/or (e,2e) reaction theory. At the experimental level EMS measurements of the valence orbitals of water, with further improved statistical accuracy will be needed as even finer details of the target molecule-ion overlap and the (e,2e) reaction theory are investigated. In the meantime the present work has considerably extended understanding of the molecular (e,2e) reaction and the use of electron momentum spec-

A.O. Bawogan et al. / EMS

of the valence orbital3 of H20 and DzO

troscopy as a sensitive diagnostic tool in molecular quantum mechanics by the detailed study of the shapes and magnitudes of orbital momentum profiles. The present work clearly demonstrates the effects of electron correlation and relaxation and the fact that good agreement is only obtained between theory and experiment if the ion-neutral overlap (i.e. the electronic structure factor) is computed with sufficient accuracy. The good overall quantitative agreement now obtained between experiment and theory for the valence orbitals of H,O also indicates the general suitability of the PWIA for the study of small molecules by EMS at impact energies of 1200 eV using the symmetric non-coplanar geometry at 8 = 45 O. However, small discrepancies between the PWIA treatment and experiment suggest that a careful investigation of distortion effects by molecular targets and in particular highly polar molecules would be informative. The present studies also demonstrate clearly the need to consider carefully the low-momentum (p) portion of molecular wavefunctions and the importance in many cases of electron correlation in the valence orbitals of small molecules such as H,O. These effects will have to be taken into account if molecular wavefunctions are to be used for highly accurate investigation of problems of bonding and the calculation of charge, spin and momentum distributions.

Acknowledgement Financial support for this work was provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the National Science Foundation (USA). We gratefully acknowledge the receipt of a University of British Columbia Graduate Fellowship (AOB) and a Canada Council Killam Research Fellowship 1984-86 (CEB).

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and E. Weigold,

Phys. Rept.

27 C (1976)

41

[2] E. Weigold and I.E. McCarthy, Advan. At. Mol. Phys. 14 (1978) 127. [3] K.T. Leung and C.E. Brion, J. Electron Spectry. 35 (1985) 327. [4] C.E. Brion, Intern. J. Quantum Chem. 2 (1986) 1397. 151 K.T. Leung and C.E. Brion, J. Am. Chem. Sot. 106 (1984) 5859. [61 K.T. Leung and C.E. Brion, Chem. Phys. 82 (1983) 113. 171 S.T. Hood, A. Hamnett and C.E. Brion, J. Electron Spectry. 11 (1977) 205. PI A.J. Dixon, S. Dey, I.E. McCarthy, E. Weigold and G.R.J. Williams, Chem. Phys. 21 (1977) 81. C.E. Brion, J.P.D. [91 R. Cambi, G. Ciullo, A. Sgamelloti, Cook, I.E. McCarthy and E. Weigold, Chem. Phys. 91 (1984) 373; 98 (1985) 185. L.Y. Lee, K.T. Leung and C.E. Brion, IlO1 A.O. Bawagan, Chem. Phys. 99 (1985) 367. [Ill C.E. Brion, S.T. Hood, I.H. Suzuki, E. Weigold and G.R.J. Williams, J. Electron. Spectry. 21 (1980) 71. and C.E. Brion, Chem. Phys. WI S.T. Hood, A. Hamnett Letters 39 (1976) 252. J.H. Moore, M.A. Coplan P31 J.A. Tossell, S.M. Lederman, and D.J. Chomay, J. Am. Chem. Sot. 106 (1984) 976. R. Muller-Fiedler, C.E. Brion, E.R. [I41 A.O. Bawagan, Davidson and D. Feller, to be published. [I51 K.T. Leung and C.E. Brion, Chem. Phys. 82 (1983) 87. S.T. Hood and C.E. Brion, J. Electron. WI A. Hamnett, Spectry. 11 (1977) 268. 1171 J.P.D. Cook, C.E. Brion and A. Hamnett, Chem. Phys. 45 (1980) 1. WI C.E. Brion, I.E. McCarthy, I.H. Suzuki, E. Weigold, G.R.J. Williams, K.L. Bedford, A.B. Kunz and E. Weidman, J. Electron Spectry. 27 (1982) 83. 1191 C. French, A.O. Bawagan, C.E. Brion, E.R. Davidson and D. Feller, to be published. PO1 B. Lohmann and E. Weigold, Phys. Letters A 86 (1981) 139. WI J. Mitroy, K. Amos and I. Morrison, J. Phys. B 17 (1984) 1659. J. Mitroy and E. Weigold, P21 J.P.D. Cook, I.E. McCarthy, Phys. Rev. A 33 (1986) 211. v31 L. Frost and E. Weigold, J. Phys. B 15 (1982) 2531. ~241 E. Weigold, I.E. McCarthy, A.J. Dixon and S. Dey, Chem. Phys. Letters 47 (1977) 209. 1251I.E. McCarthy and E. Weigold, Phys. Rev. A 31 (1985) 160. A. Duguet and C. Dal Capello, J. 1261A. Lahmann-Benani, Electron Spectry. 40 (1986) 141. and E. Weigold, Chem. 1271 G.R.J. Williams, I.E. McCarthy Phys. 22 (1977) 281. PI C.E. Brion, J.P.D. Cook, I.G. Fuss and E. Weigold, Chem. Phys. 64 (1982) 287. A.T. Stelbovics and E. [291 J.P.D. Cook, I.E. McCarthy, Weigold, J. Phys. B 17 (1984) 2339. 1301 I.E. McCarthy, J. Electron Spectry. 36 (1985) 37. J.T.N. Kimman, M. van Tilburg and [311 B. van Wingerden, F.J. de Heer, J. Phys. B 14 (1981) 2475.

42

A.O. Bawagan et al. / EMS

of the valence

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orbitals of HI0 and D,O

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