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Vibrational averaging effects on the valence-shell electron momentum distributions in H2O employing Hartree-Fock-limit wavefunctions
Volume 157, number
CHEMICAL
i,2
28 April 1989
PHYSICS LETTERS
VIBRATIONAL AVERAGING EFFECTS ON THE VALENCE-SHELL ELECTRON MOMENTUM DISTRIBUTIONS EMPLOYING HARTREE-FOCK-LIMIT WAVEFUNCTIONS
IN HZ0
K.T. LEUNG
’ ofChemistry,
Department
University
J.G. SHEEHY and P.W. LANGHOFF Department
Received
ofChemistry,
lndrana
Waterloo, Ontario.
of Waterloo,
University.
Canada N2L 3GI
* Bloomrngton,
I4 April 1988; in final form 7 February
IN 47405,
USA
1989
Vibrational averaging effects on electron momentum distributions in the valence shell of H,O are calculated employing Hartree-Fock-limit electronic wavefunctions and an accurate empirical ground-state potential energy surface. The effects arc generally small, lending support to the widespread practice of evaluating theoretical electron momentum distributionsat fixedequilibrium geometries, in accordance with summations over unresolved final vibrational states in the vertical electronic Franck-Condon approximation. Vibrational corrections to the angle-averaged 3a, and 1b2 orbital momentum distributions in HZO, slightly reduce existing discrepancies between calculations and previously reported (e, 2e) experiments for small values of target electron momentum. This suggests that vibrational averaging should generally be incorporated in refined computational studies of electron momentum distributions in polyatomic molecules, and should be combined with the generally more significant effects of finite experimental momentum resolution when comparisons are made with measured values derived from (e, 2e) spectroscopy.
1. Introduction Continuing refinements in (e, 2e) spectroscopy have provided an increasingly precise tool for the study of [ 11. Improvements in experimental momentum resolution and in signal-to-noise perelectronic wavefunctions formance, in particular, have led to more accurate measurements of the spherically averaged electron momentum distributions of atoms and molecules for comparison with ab initio ground-state wavefunctions. Detailed descriptions of the experimental method, theory, and application of (e, 2e) spectroscopy are given elsewhere [l-3], and a recent bibliography summarizes reported (e, 2e) studies of electron momentum distributions in over 50 atoms and molecules [ 41. In this Letter, studies are reported of the validity of the vertical electronic, Frank-Condon approximation commonly employed in the interpretation of (e, 2e) electron scattering experiments. Experimental investigations employing isotope substitution in HZ/D2 [ 51, and more recently in HzO/D20 [ 61, suggest mass effects are small in these cases, lending support to the usual practice of ignoring vibrational averaging in computational studies. In view of the apparent absence of confirming calculations, however, and of the possibility that vibrational effects may be important in accounting for continuing small discrepancies between experiments and calculations of electron momentum distributions in molecules [ 6,7 1, appropriate computational studies are clearly in order. Accordingly, calculations are reported here in the vibrational-averaging approximation for the valence-shell electron momentum distributions in H,O, comparisons are made with results obtained in the ver’ Work supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. ’ Work supported in part by grants from the National Science Foundation (CHE86- 14344) and by the US Department FC05.85ER250000).
0 009-26 14189 / $ 03.50 0 Elsevier Science Publishers (North-Holland
Physics Publishing
Division )
B.V.
of Energy (DE-
135
Volume
157, number I ,2
CHEMICAL
PHYSICS
LETTERS
28 April 1989
tical electronic approximation, and the relevance of the present studies to the interpretation of experimental (e, 2e) measurements is commented on. Equilibrium and turning-point geometries adopted in the present study are obtained from an accurate empirical potential-energy surface devised to reproduce available spectroscopic data in Hz0 [ 81. Ground-state electronic wavefunctions are calculated at these geometries employing a basis set previously found to provide Hartree-Fock-limit expectation values for selected electronic properties at the equilibrium geometry [ 91. A simple quadrature at anharmonic turning points determined in the directions of the equilibrium normal-mode coordinates is employed to approximate the vibrational average of the relevant electron momentum distributions. Although previously reported investigations of vibrational effects on certain electronic properties of Hz0 indicate that symmetric modes provide the major corrections to equilibrium geometry values in the cases studied [ 10-21, attention is addressed here to all three (symmetric stretch, asymmetric stretch, and bend) vibrations in the absence of previous reports of their effects on electron momentum distributions in this compound. The calculated angle-averaged electron momentum distributions evaluated at the turning-point geometries arc found to differ noticeably from the corresponding equilibrium values, particularly for the 2a,, 3a,, and lb* orbitals. There are smaller differences from the equilibrium or vertical electronic, Franck-Condon distributions for most values of momentum when the vibrational averages are performed. However, the effects of averaging are seen to persist for the 3a, and 1b2 orbitals for small values of target electron momentum, suggesting that vibrational averaging should generally be incorporated in precise calculations of electron momentum distributions, and should be combined with the generally more significant consequences of finite experimental momentum resolution when comparisons are made with measured values derived from (e, 2e) spectroscopy. 2. Theoretical
development
Use of (e, 2e) spectroscopy as a probe of electronic wavefunctions relies in part upon simplifications of the theoretical expression for the (e, 2e) scattering cross section employed in the interpretation of the measured data. These simplifications involve the use of (a) the conventional Born-Oppenheimer approximation for the initial state and final ionic states of the molecule, (b) the plane-wave impulse approximation for the scattering kinematics in the (e, 2e) reaction, (c) the Hartree-Fock approximation for the target-state electronic wavefunction, and (d) the single-hole or Koopmans approximation to the final ionic state [ 2,3]. The appropriate triply differential cross section d%/dE, d8, d& can be written in the general form [ 21 d301d.5 dn, dQ2 = (2n: I4 zav (PI PJPO 1 I (
Y+‘#-
1)
I T(E)
I @o ‘Y,(J’O
> I2
.
(1)
Here, T(E) is a transition operator describing the (e, 2e) reaction, 1, represents an average over the initial degenerate states and a sum over the unresolved final states, ~j and pi are the wavefunctions and momenta, respectively, of the incident ( i= 0) and outgoing (i= 1, 2) electrons, and !Pg(N) and ‘$(N- 1) are the wavefunctions of the neutral target-state molecule and the fth ionic state, respectively [ 2,3]. The four approximations (a) to (d) indicated above reduce the triply differential cross section of eq. ( 1) to a product of the Mott scattering cross-section, (7M01r,and a rotational and vibrational integral over the Fourier transform of a canonical Hartree-Fock orbital [ 2,3]. Specifically d”aldE,
dQ, dQz-+(2x)4(~,Pzl~,n)
Cav I <9141W”x.f”’ I T(E) I~%-“x:~‘>
-(2x)4(~I~rl~oMMotr CavI (exp(ip-4V”xY
1
I ~PP>12,
(2a) (2b)
2 --t (2~)4(PfP21Po)R.ml
dQ i
+ (2~)4(P,PrlPohl~*l
136
s
&
IX:“‘(q) I* 1 ~~@(P; 4) .I
dq
Ix:“(q)I”l@1-(P;q)12 7
s
da
5
1
(2c) (2d)
Volume 157, number I,2
CHEMICALPHYSICSLETTERS
28 April 1989
where p ( =pO- (p, +pz) ) is the momentum of the bound electron prior to the knock-out, p(e’) and xc”) (q) are the molecular electronic and vibrational wavefunctions, respectively, and a, is the amplitude of the jth single-hole configuration in the fth ionic state, with Icz~~ ’ providing the corresponding spectroscopic factor. Note that two-hole, one-particle and higher-order terms do not contribute to eq. (2~) when the target-state canonical Fock orbitals are employed to enumerate the hole-particle contributions to the ionic states [ 131. The momentum-space orbital $J&; q) at the molecular geometry q is related to the canonical Hartree-Fock orbital &(r; q) by the usual Fourier transform (@,(p; q) =Jdrexp( -ip+r) &(r; q) ). The validity of the Born-Oppenheimer approximation employed in eq. (2a) is well established and needs no discussion here. The plane-wave impulse approximation in eq. (2b) has been examined extensively in the early theoretical development of (e, 2e) spectroscopy, and is found to provide an accurate approximation to the scattering part of the (e, 2e) cross section for all cases studied to date in the momentum range z 0 to 1.5 au for impact energies above M 1 keV [ 3,14,15 1. The use of a target-state Hartree-Fock wavefunction in eq. (2~) is valid in Hz0 [ 6,161, and is generally valid in the absence of significant ground-state electronic correlation, even in the presence of the ionic-state configuration mixing common in second-row molecules (e.g. C!$ and OCS) [ 4,17 1. Finally, the single-hole state or Koopmans approximation of eq. (2d) is appropriate for the main line lb; ’, 3a, ’ and lb, ’ ionic states in Hz0 [ 161. Moreover, although it is well known that ionic-state configuration mixing splits the inner-valence 2a; ’ line in Hz0 [ 6,161, the result of eq. (2d) in this case should provide a good approximation to the measured momentum distribution corresponding to integration over the appropriate inner-valence ejected-electron kinetic-energy band in the absence of other contributing factors [ 6,7]. In the vertical electronic, Franck-Condon approximation, the density Ix:“) (q) 1’ in the vibrational integral (Jdq) of eq. (2d) is replaced by a delta function centered at the equilibrium geometry (q=qe), whereas this integral is evaluated explicitly in the vibrational averaging approximation adopted here, incorporating the change of Igl&; q) I’ with molecular geometry. Since recent studies show that there are significant improvements between calculations and (e, 2e) experiments in first-row hydrides and in second-row molecules when HartreeFock limit wavefunctions are used [ 6,17 1, it is important to employ a high-quality Hat-tree-Fock wavefunction to evaluate the vibrational integral.
3. Computations The empirical potential energy surface used in the present study of vibrational averaging effects on the electron momentum distributions in Hz0 correctly reproduces the energies of a significant number of vibrational states, as well as the experimental equilibrium geometry [ 81. Six sets of nuclear coordinates, corresponding to anharmonic turning-point geometries in the directions of the three equilibrium normal-mode vibrations and to the equilibrium geometry on this potential surface, are used in the study. The turning-point geometries in the equilibrium normal-mode directions are reached when the associated vibrational wavefunctions are reduced to values that provide relative quadrature weights of I/e. In this way anharmonic effects are approximately incorporated in the development without performing the somewhat cumbersome expansion of electronic properties in high powers of the normal-model coordinates [ 10,111. Hartree-Fock wavefunctions are constructed at the six molecular geometries using a basis set of 126 contracted Gaussian functions [ 9 1. This basis set gives a wavefunction at the equilibrium geometry that is judged to be very close to the Hartree-Fock limit [9], and, consequently, is satisfactory for present purposes. The relevant geometries and the total energies of the corresponding SCF wavefunctions are given in table 1. Note that the asymmetric stretching motion has only one distinct turning-point geometry, whereas the symmetric stretch and bend motions have two distinct turning-point configurations. Examination of the potential curves
137
Volume 157, number Table I Molecular geometries averaging calculations
1,2
CHEMICAL
PHYSICS LETTERS
and corresponding Hartree-Fock energies on the ground-state ofelectron momentum distributions a)
Molecular
configuration
equilibrium di symmetric stretch, v, compressed stretched bend, p2 bend in bend out asymmetric stretch, v, HI position Hz position
‘)
Hydrogen-atom (11.4304,
coordinates
1.1071)
28 April I989
H,O potential
(a,) Cl
surface employed
Total energy (au) -76.0672
( + 1.3389, I .0X3) (rk 1.5376, 1.1901)
- 76.0629 -76.0534
(+ 1.3119, 1.2453)
-76.0627 -76.0637
(i 1.5489, 0.9342) (+1.5277, (-1.3332,
1.1824) 1.0319)
in vibrational
‘)
-76.0586
.‘I Equilibrium and turning-point geometries obtained from the potential surface ofref. [8]. The turning-point geometries aredefined by displacements in the equilibrium normal-mode directions ofmagnitudes which reduce the appropriate vibrational wavefunction probabilities to zz 1/P of their equilibrium geometry values. h1 Note that the symmetric stretch and bend motions have both inner and outer turning-point geometries, whereas the asymmetric stretch does not. I’ Hydrogen-atom coordinates (.v, z) at the potential minimum and the turning points of the indicated normal-mode vibrations. The oxygen atom is at the origin, and all x coordinates are 0. ” The energy minimum position on the empirical potential surface employed coincides with the experimental equilibrium geometry [gl. ‘) The Hartree-Fock
limit at this point is judged to be - 76.0675 au
[ 91.
(not shown) along the equilibrium normal-mode directions reveals the presence of anharmonic behavior in the vicinities of the turning-point geometries_ The angle-averaged electron momentum distributions of the valence-shell orbitals (2a,, 1bz, 3a,, 1b, ) in H20, evaluated at the six molecular geometries employed in the present study, are shown in fig.1. Several observations are appropriate. First, the lb, orbital results are seen to be largely invariant to changes in the molecular geometry up to the turning points, as may be anticipated from the out-of-plane, lone-pair nature of this orbital, Second, significant differences are evident between the turning-point electron momentum distributions and the corresponding potential minimum values for the other three orbitals (3a,, lb2 and 2a, ). These differences are largely, but not entirely, in the amplitudes of the calculated distributions. The shapes of the momentum distributions remain relatively unaltered with change in geometry from those evaluated at the potential minimum. There are, however, noticeable differences between the turning-point and equilibrium momentum distributions for small values of momentum in all three orbitals (3a,, lb,, 2a,; see fig. 1) Vibrational averages of the angle-averaged momentum distributions in Hz0 are calculated employing the results of fig. 1 and a six-point quadrature to the vibrational integral (eq. (2~) ). As indicated above, tumingpoint weights of 1/e (2/e for the asymmetric stretch), which define the extent of motion in the equilibrium normal-mode directidns, are adopted in accordance with the vibrational wavefunctions at these points. Fig. 2 compares the vibrationally averaged orbital electron momentum distributions with those at the equilibrium geometry for the four valence orbitals in H20, reported in the form of percentage changes from the equilibrium values. Very small differences are observed between the vibrationally averaged and equilibrium electron momentum distributions for the 1b, and 2a, orbitals. Vibrational motion appears to affect the momentum distribution of the 2a, orbital equally at opposite turning points (fig. la), causing cancellation of the effect in the vibrationally averaged result, whereas the I b, orbital is largely invariant to any changes in molecular geometry (fig. Id). The momentum distributions of the two bonding orbitals (3a, and lb*), however, show small but more significant vibrational effects in the low-momentum region (below 0.5 au). Moreover, the small corrections due to vibrational motion shown in fig. 2 are in directions that tend to improve agreement with the 138
Volume 157, number
1,2
CHEMICAL
28 April 1989
PHYSICS LETTERS
m 0.26
2 9
H,O
d
2
v
EQVIL BEND OUT BEND IN STR IN STR OUT
----~~~~~ .-.--.-.-._ ----------
0.20
;
1
j;/-.&
ASYMM
cl.15
o.ot, 0.6
I.0
2.0
1.5
MOMENTUM 0.07
,
,
S’S,
I
,
I
.n
I
,
0.0
I
!
I
I
,.y
t
‘\ ’ ?,’ _’
MOMENTUM
H,O
EQVIL BEND OUT BEND IN STR IN
----------
ASYMM
STR
OUT
a.0
MOMENTUM
(au.)
MOMENTUM
(au.)
2.5
3.0
C
3a,
-
---__ “._...__._ -.. .-.-.--.
,
',i‘,
:j ;i
-
3.0
2.5
(a.~.) ,
lb,
-----, _._._._._..
PQVIL BEND OUT BEND IN STR IN
----------
STR OUT ASYMM
(a-u.)
Fig. 1. Spherically averaged valence-shell electron momentum distributions in H,O calculated using near Hartree-Fock-limit wavefunctions at the equilibrium and turning-point geometries of table 1, as indicated: (a) 2a, orbital, (b) lb1 orbital, (c) 3a, orbital, and (d) 1b , orbital.
measured (e, 2e) values, although they are small in comparison finite experimental momentum resolution [ 6,7].
with the generally more significant
effects of
4. Concluding remarks The magnitudes of vibrational averaging effects on the electron momentum distributions of the valence-shell orbitals of I-I,0 are reported. Near Hartree-Fock-limit wavefunctions are used to evaluate the orbital momentum distributions at the equilibrium and turning-point geometries obtained from an accurate empirical 139
Volume 157, number I,2
-5.0
0.0
’
’
’
0.5
CHEMICAL
’ 1.0
’
1
1.5
MOMENTUM
’
’ 2.0
’
I
2.5
(a-u.)
’ 3.0
PHYSICS LETTERS
28 April 1989
Fig. 2. Spherically averaged electron momentum distributions for the valence-shell orbitals in H20 obtained from vibrational-avwaging calculations employing eq. (2d), as described in the text, expressed as percentage deviations from the corresponding equilibrium geometry or vertical electronic, Franck-Condon values.
potential energy surface. The turning-point geometries adopted are in the directions of the equilibrium normalmode displacements, but incorporate anharmonic effects (the magnitudes of the stretch and bend motions are extended until the associated vibrational wavefunctions in these directions provide relative quadrature weights of 1/e). These conditions serve to define the turning-point geometries adopted in the present study. A six-point quadrature employed in the evaluation of the vibrational averaging integral over the vibrational motion provides the desired angle-averaged orbital electron momentum distributions in H20. The calculated vibrational averaging effects on the angle-averaged electron momentum distributions are found to be small, despite the more significant dependence found on molecular geometry shown in fig. 1. This circumstance is largely a consequence of the rapid drop-off in the amplitude of the vibrational wavefunctions away from equilibrium geometry, and of partial cancellations of contributions to the vibrational integral from opposite turning points in the symmetric vibrations. In the harmonic approximation, which is not adopted in the present study, such cancellations would be complete for the symmetric normal-mode vibrations. Small corrections due to vibrational averaging in the momentum distributions of the 3a, and 1bz orbitals in the lowmomentum region shown in fig. 2 appear to reduce the discrepancy between experiment and theory, suggesting that vibrational averaging can have an important, albeit small, effect in this electron momentum region. The small vibrational averaging effects observed in this work are consistent with theoretical studies on vibrational corrections to other electronic properties of I&O, and lend further support to the practice of comparing experimental electron momentum distributions with theoretical values calculated at equilibrium geometries. In view of the noticeable differences between the turning-point and equilibrium momentum distributions of the 3a,, lb? and 2a, orbitals in H20, of the increase in amplitudes at low momentum, in particular, and of the possibility of the non-cancellation of such effects at opposite turning points in other compounds, vibrational averaging should generally be incorporated in refined computational studies of electron momentum distributions in polyatomic molecules. These considerations should also be combined with the more significant effects of finite experimental momentum resolution when comparisons are made with results derived from (e, 2e) spectroscopy measurements.
Acknowledgement Acknowledgement 140
is made to the Natural
Sciences and Engineering
Research Council of Canada, to the Na-
Volume 157, number
1.2
CHEMICAL
PHYSICS LETTERS
28 April 1989
tional Science Foundation, and to the Department of Energy for partial support of these studies. One of us (KTL) thanks his collaborators for their kind hospitality during his visit to Bloomington, Indiana, in the fall of 1984. The use of facilities of the National Center for Supercomputing Applications is gratefully acknowledged.
References [ 1] [2] [3] [4]
C.E. Brion, Intern. J. Quantum Chem. 29 (1986) 1397. I.E. McCarthy and E. Weigold, Phys. Rept. 27 (1976) 275. I.E. McCarthy and E. Weigold, Contemp. Phys. 244 (1983) 163. K.T. Lcung and C.E. Brian, J. Electron Spectry. 35 ( 1985) 327. [ 51S.Dey, I.E. McCarthy, P-J-0. Teubner and E. Weigold, Phys. Rev. Letters 34 ( 1975) 782. [6] A.O. Bawagan, C.E. Brion, E.R. Davidson and D. Feller, Chem. Phys. 1 I3 (1987) 19. [7] A.O. Bawagan, L.Y. Lee, K.T. Leung and C.E. Brion, Chem. Phys. 99 ( 1985) 367. [ 81J.N. Murrell, S. Carter, SC. Farantos, P. Huxley and A.J.C. Varandas, Molecular potential energy functions (Wiley, New York, 1984) p. 133. [9] D. Feller, CM. Boyle and E.R. Davidson, J. Chem. Phys. 86 (1987) 3424. [IO] C.W. Kern and R.L. Matcha, J. Chem. Phys. 49 (1968) 2081. [1I]W.C.ErmlerandC.W.Kem,J.Chem.Phys.55(1971)4851. [ 121 M. Breitenstein, R.J. Mawhorter, H. Meyer and A. Schweig, Mol. Phys. 57 ( 1986) 8 I. [ 131 P.W. Langhoff, S.R. Langhoff, T.N. Rescigno, J.S. Schirmer, L.S, Cederbaum, W. Domcke and W. von Niessen, Chem. Phys. 58 (1981) 71. [ 141 G.R.J. Williams, I.E. McCarthy and E. Weigold, Chem. Phys. 22 (1977) 281. [ 151I.E. McCarthy, in: Coherence and correlation in atomic collisions, eds. H. Kleinpoppen and J.F. Williams (Plenum Press, New York, 1980) p. 1. [ 161 G.H.F. Diercksen, W.P. Kraemer, T.N. Rescigno, C.F. Bender, B.V. McKay, S.R. Langhoff and P. W. Langhoff, J. Chem. Phys. 76 (1982) 1043. [ 171 K.T. Leung, C.E. Brion, B.W. Fatyga and P.W. Langhoff, Chem. Phys. 96 (I 985) 227.
141
CHEMICAL
i,2
28 April 1989
PHYSICS LETTERS
VIBRATIONAL AVERAGING EFFECTS ON THE VALENCE-SHELL ELECTRON MOMENTUM DISTRIBUTIONS EMPLOYING HARTREE-FOCK-LIMIT WAVEFUNCTIONS
IN HZ0
K.T. LEUNG
’ ofChemistry,
Department
University
J.G. SHEEHY and P.W. LANGHOFF Department
Received
ofChemistry,
lndrana
Waterloo, Ontario.
of Waterloo,
University.
Canada N2L 3GI
* Bloomrngton,
I4 April 1988; in final form 7 February
IN 47405,
USA
1989
Vibrational averaging effects on electron momentum distributions in the valence shell of H,O are calculated employing Hartree-Fock-limit electronic wavefunctions and an accurate empirical ground-state potential energy surface. The effects arc generally small, lending support to the widespread practice of evaluating theoretical electron momentum distributionsat fixedequilibrium geometries, in accordance with summations over unresolved final vibrational states in the vertical electronic Franck-Condon approximation. Vibrational corrections to the angle-averaged 3a, and 1b2 orbital momentum distributions in HZO, slightly reduce existing discrepancies between calculations and previously reported (e, 2e) experiments for small values of target electron momentum. This suggests that vibrational averaging should generally be incorporated in refined computational studies of electron momentum distributions in polyatomic molecules, and should be combined with the generally more significant effects of finite experimental momentum resolution when comparisons are made with measured values derived from (e, 2e) spectroscopy.
1. Introduction Continuing refinements in (e, 2e) spectroscopy have provided an increasingly precise tool for the study of [ 11. Improvements in experimental momentum resolution and in signal-to-noise perelectronic wavefunctions formance, in particular, have led to more accurate measurements of the spherically averaged electron momentum distributions of atoms and molecules for comparison with ab initio ground-state wavefunctions. Detailed descriptions of the experimental method, theory, and application of (e, 2e) spectroscopy are given elsewhere [l-3], and a recent bibliography summarizes reported (e, 2e) studies of electron momentum distributions in over 50 atoms and molecules [ 41. In this Letter, studies are reported of the validity of the vertical electronic, Frank-Condon approximation commonly employed in the interpretation of (e, 2e) electron scattering experiments. Experimental investigations employing isotope substitution in HZ/D2 [ 51, and more recently in HzO/D20 [ 61, suggest mass effects are small in these cases, lending support to the usual practice of ignoring vibrational averaging in computational studies. In view of the apparent absence of confirming calculations, however, and of the possibility that vibrational effects may be important in accounting for continuing small discrepancies between experiments and calculations of electron momentum distributions in molecules [ 6,7 1, appropriate computational studies are clearly in order. Accordingly, calculations are reported here in the vibrational-averaging approximation for the valence-shell electron momentum distributions in H,O, comparisons are made with results obtained in the ver’ Work supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. ’ Work supported in part by grants from the National Science Foundation (CHE86- 14344) and by the US Department FC05.85ER250000).
0 009-26 14189 / $ 03.50 0 Elsevier Science Publishers (North-Holland
Physics Publishing
Division )
B.V.
of Energy (DE-
135
Volume
157, number I ,2
CHEMICAL
PHYSICS
LETTERS
28 April 1989
tical electronic approximation, and the relevance of the present studies to the interpretation of experimental (e, 2e) measurements is commented on. Equilibrium and turning-point geometries adopted in the present study are obtained from an accurate empirical potential-energy surface devised to reproduce available spectroscopic data in Hz0 [ 81. Ground-state electronic wavefunctions are calculated at these geometries employing a basis set previously found to provide Hartree-Fock-limit expectation values for selected electronic properties at the equilibrium geometry [ 91. A simple quadrature at anharmonic turning points determined in the directions of the equilibrium normal-mode coordinates is employed to approximate the vibrational average of the relevant electron momentum distributions. Although previously reported investigations of vibrational effects on certain electronic properties of Hz0 indicate that symmetric modes provide the major corrections to equilibrium geometry values in the cases studied [ 10-21, attention is addressed here to all three (symmetric stretch, asymmetric stretch, and bend) vibrations in the absence of previous reports of their effects on electron momentum distributions in this compound. The calculated angle-averaged electron momentum distributions evaluated at the turning-point geometries arc found to differ noticeably from the corresponding equilibrium values, particularly for the 2a,, 3a,, and lb* orbitals. There are smaller differences from the equilibrium or vertical electronic, Franck-Condon distributions for most values of momentum when the vibrational averages are performed. However, the effects of averaging are seen to persist for the 3a, and 1b2 orbitals for small values of target electron momentum, suggesting that vibrational averaging should generally be incorporated in precise calculations of electron momentum distributions, and should be combined with the generally more significant consequences of finite experimental momentum resolution when comparisons are made with measured values derived from (e, 2e) spectroscopy. 2. Theoretical
development
Use of (e, 2e) spectroscopy as a probe of electronic wavefunctions relies in part upon simplifications of the theoretical expression for the (e, 2e) scattering cross section employed in the interpretation of the measured data. These simplifications involve the use of (a) the conventional Born-Oppenheimer approximation for the initial state and final ionic states of the molecule, (b) the plane-wave impulse approximation for the scattering kinematics in the (e, 2e) reaction, (c) the Hartree-Fock approximation for the target-state electronic wavefunction, and (d) the single-hole or Koopmans approximation to the final ionic state [ 2,3]. The appropriate triply differential cross section d%/dE, d8, d& can be written in the general form [ 21 d301d.5 dn, dQ2 = (2n: I4 zav (PI PJPO 1 I (
Y+‘#-
1)
I T(E)
I @o ‘Y,(J’O
> I2
.
(1)
Here, T(E) is a transition operator describing the (e, 2e) reaction, 1, represents an average over the initial degenerate states and a sum over the unresolved final states, ~j and pi are the wavefunctions and momenta, respectively, of the incident ( i= 0) and outgoing (i= 1, 2) electrons, and !Pg(N) and ‘$(N- 1) are the wavefunctions of the neutral target-state molecule and the fth ionic state, respectively [ 2,3]. The four approximations (a) to (d) indicated above reduce the triply differential cross section of eq. ( 1) to a product of the Mott scattering cross-section, (7M01r,and a rotational and vibrational integral over the Fourier transform of a canonical Hartree-Fock orbital [ 2,3]. Specifically d”aldE,
dQ, dQz-+(2x)4(~,Pzl~,n)
Cav I <9141W”x.f”’ I T(E) I~%-“x:~‘>
-(2x)4(~I~rl~oMMotr CavI (exp(ip-4V”xY
1
I ~PP>12,
(2a) (2b)
2 --t (2~)4(PfP21Po)R.ml
dQ i
+ (2~)4(P,PrlPohl~*l
136
s
&
IX:“‘(q) I* 1 ~~@(P; 4) .I
dq
Ix:“(q)I”l@1-(P;q)12 7
s
da
5
1
(2c) (2d)
Volume 157, number I,2
CHEMICALPHYSICSLETTERS
28 April 1989
where p ( =pO- (p, +pz) ) is the momentum of the bound electron prior to the knock-out, p(e’) and xc”) (q) are the molecular electronic and vibrational wavefunctions, respectively, and a, is the amplitude of the jth single-hole configuration in the fth ionic state, with Icz~~ ’ providing the corresponding spectroscopic factor. Note that two-hole, one-particle and higher-order terms do not contribute to eq. (2~) when the target-state canonical Fock orbitals are employed to enumerate the hole-particle contributions to the ionic states [ 131. The momentum-space orbital $J&; q) at the molecular geometry q is related to the canonical Hartree-Fock orbital &(r; q) by the usual Fourier transform (@,(p; q) =Jdrexp( -ip+r) &(r; q) ). The validity of the Born-Oppenheimer approximation employed in eq. (2a) is well established and needs no discussion here. The plane-wave impulse approximation in eq. (2b) has been examined extensively in the early theoretical development of (e, 2e) spectroscopy, and is found to provide an accurate approximation to the scattering part of the (e, 2e) cross section for all cases studied to date in the momentum range z 0 to 1.5 au for impact energies above M 1 keV [ 3,14,15 1. The use of a target-state Hartree-Fock wavefunction in eq. (2~) is valid in Hz0 [ 6,161, and is generally valid in the absence of significant ground-state electronic correlation, even in the presence of the ionic-state configuration mixing common in second-row molecules (e.g. C!$ and OCS) [ 4,17 1. Finally, the single-hole state or Koopmans approximation of eq. (2d) is appropriate for the main line lb; ’, 3a, ’ and lb, ’ ionic states in Hz0 [ 161. Moreover, although it is well known that ionic-state configuration mixing splits the inner-valence 2a; ’ line in Hz0 [ 6,161, the result of eq. (2d) in this case should provide a good approximation to the measured momentum distribution corresponding to integration over the appropriate inner-valence ejected-electron kinetic-energy band in the absence of other contributing factors [ 6,7]. In the vertical electronic, Franck-Condon approximation, the density Ix:“) (q) 1’ in the vibrational integral (Jdq) of eq. (2d) is replaced by a delta function centered at the equilibrium geometry (q=qe), whereas this integral is evaluated explicitly in the vibrational averaging approximation adopted here, incorporating the change of Igl&; q) I’ with molecular geometry. Since recent studies show that there are significant improvements between calculations and (e, 2e) experiments in first-row hydrides and in second-row molecules when HartreeFock limit wavefunctions are used [ 6,17 1, it is important to employ a high-quality Hat-tree-Fock wavefunction to evaluate the vibrational integral.
3. Computations The empirical potential energy surface used in the present study of vibrational averaging effects on the electron momentum distributions in Hz0 correctly reproduces the energies of a significant number of vibrational states, as well as the experimental equilibrium geometry [ 81. Six sets of nuclear coordinates, corresponding to anharmonic turning-point geometries in the directions of the three equilibrium normal-mode vibrations and to the equilibrium geometry on this potential surface, are used in the study. The turning-point geometries in the equilibrium normal-mode directions are reached when the associated vibrational wavefunctions are reduced to values that provide relative quadrature weights of I/e. In this way anharmonic effects are approximately incorporated in the development without performing the somewhat cumbersome expansion of electronic properties in high powers of the normal-model coordinates [ 10,111. Hartree-Fock wavefunctions are constructed at the six molecular geometries using a basis set of 126 contracted Gaussian functions [ 9 1. This basis set gives a wavefunction at the equilibrium geometry that is judged to be very close to the Hartree-Fock limit [9], and, consequently, is satisfactory for present purposes. The relevant geometries and the total energies of the corresponding SCF wavefunctions are given in table 1. Note that the asymmetric stretching motion has only one distinct turning-point geometry, whereas the symmetric stretch and bend motions have two distinct turning-point configurations. Examination of the potential curves
137
Volume 157, number Table I Molecular geometries averaging calculations
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CHEMICAL
PHYSICS LETTERS
and corresponding Hartree-Fock energies on the ground-state ofelectron momentum distributions a)
Molecular
configuration
equilibrium di symmetric stretch, v, compressed stretched bend, p2 bend in bend out asymmetric stretch, v, HI position Hz position
‘)
Hydrogen-atom (11.4304,
coordinates
1.1071)
28 April I989
H,O potential
(a,) Cl
surface employed
Total energy (au) -76.0672
( + 1.3389, I .0X3) (rk 1.5376, 1.1901)
- 76.0629 -76.0534
(+ 1.3119, 1.2453)
-76.0627 -76.0637
(i 1.5489, 0.9342) (+1.5277, (-1.3332,
1.1824) 1.0319)
in vibrational
‘)
-76.0586
.‘I Equilibrium and turning-point geometries obtained from the potential surface ofref. [8]. The turning-point geometries aredefined by displacements in the equilibrium normal-mode directions ofmagnitudes which reduce the appropriate vibrational wavefunction probabilities to zz 1/P of their equilibrium geometry values. h1 Note that the symmetric stretch and bend motions have both inner and outer turning-point geometries, whereas the asymmetric stretch does not. I’ Hydrogen-atom coordinates (.v, z) at the potential minimum and the turning points of the indicated normal-mode vibrations. The oxygen atom is at the origin, and all x coordinates are 0. ” The energy minimum position on the empirical potential surface employed coincides with the experimental equilibrium geometry [gl. ‘) The Hartree-Fock
limit at this point is judged to be - 76.0675 au
[ 91.
(not shown) along the equilibrium normal-mode directions reveals the presence of anharmonic behavior in the vicinities of the turning-point geometries_ The angle-averaged electron momentum distributions of the valence-shell orbitals (2a,, 1bz, 3a,, 1b, ) in H20, evaluated at the six molecular geometries employed in the present study, are shown in fig.1. Several observations are appropriate. First, the lb, orbital results are seen to be largely invariant to changes in the molecular geometry up to the turning points, as may be anticipated from the out-of-plane, lone-pair nature of this orbital, Second, significant differences are evident between the turning-point electron momentum distributions and the corresponding potential minimum values for the other three orbitals (3a,, lb2 and 2a, ). These differences are largely, but not entirely, in the amplitudes of the calculated distributions. The shapes of the momentum distributions remain relatively unaltered with change in geometry from those evaluated at the potential minimum. There are, however, noticeable differences between the turning-point and equilibrium momentum distributions for small values of momentum in all three orbitals (3a,, lb,, 2a,; see fig. 1) Vibrational averages of the angle-averaged momentum distributions in Hz0 are calculated employing the results of fig. 1 and a six-point quadrature to the vibrational integral (eq. (2~) ). As indicated above, tumingpoint weights of 1/e (2/e for the asymmetric stretch), which define the extent of motion in the equilibrium normal-mode directidns, are adopted in accordance with the vibrational wavefunctions at these points. Fig. 2 compares the vibrationally averaged orbital electron momentum distributions with those at the equilibrium geometry for the four valence orbitals in H20, reported in the form of percentage changes from the equilibrium values. Very small differences are observed between the vibrationally averaged and equilibrium electron momentum distributions for the 1b, and 2a, orbitals. Vibrational motion appears to affect the momentum distribution of the 2a, orbital equally at opposite turning points (fig. la), causing cancellation of the effect in the vibrationally averaged result, whereas the I b, orbital is largely invariant to any changes in molecular geometry (fig. Id). The momentum distributions of the two bonding orbitals (3a, and lb*), however, show small but more significant vibrational effects in the low-momentum region (below 0.5 au). Moreover, the small corrections due to vibrational motion shown in fig. 2 are in directions that tend to improve agreement with the 138
Volume 157, number
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28 April 1989
PHYSICS LETTERS
m 0.26
2 9
H,O
d
2
v
EQVIL BEND OUT BEND IN STR IN STR OUT
----~~~~~ .-.--.-.-._ ----------
0.20
;
1
j;/-.&
ASYMM
cl.15
o.ot, 0.6
I.0
2.0
1.5
MOMENTUM 0.07
,
,
S’S,
I
,
I
.n
I
,
0.0
I
!
I
I
,.y
t
‘\ ’ ?,’ _’
MOMENTUM
H,O
EQVIL BEND OUT BEND IN STR IN
----------
ASYMM
STR
OUT
a.0
MOMENTUM
(au.)
MOMENTUM
(au.)
2.5
3.0
C
3a,
-
---__ “._...__._ -.. .-.-.--.
,
',i‘,
:j ;i
-
3.0
2.5
(a.~.) ,
lb,
-----, _._._._._..
PQVIL BEND OUT BEND IN STR IN
----------
STR OUT ASYMM
(a-u.)
Fig. 1. Spherically averaged valence-shell electron momentum distributions in H,O calculated using near Hartree-Fock-limit wavefunctions at the equilibrium and turning-point geometries of table 1, as indicated: (a) 2a, orbital, (b) lb1 orbital, (c) 3a, orbital, and (d) 1b , orbital.
measured (e, 2e) values, although they are small in comparison finite experimental momentum resolution [ 6,7].
with the generally more significant
effects of
4. Concluding remarks The magnitudes of vibrational averaging effects on the electron momentum distributions of the valence-shell orbitals of I-I,0 are reported. Near Hartree-Fock-limit wavefunctions are used to evaluate the orbital momentum distributions at the equilibrium and turning-point geometries obtained from an accurate empirical 139
Volume 157, number I,2
-5.0
0.0
’
’
’
0.5
CHEMICAL
’ 1.0
’
1
1.5
MOMENTUM
’
’ 2.0
’
I
2.5
(a-u.)
’ 3.0
PHYSICS LETTERS
28 April 1989
Fig. 2. Spherically averaged electron momentum distributions for the valence-shell orbitals in H20 obtained from vibrational-avwaging calculations employing eq. (2d), as described in the text, expressed as percentage deviations from the corresponding equilibrium geometry or vertical electronic, Franck-Condon values.
potential energy surface. The turning-point geometries adopted are in the directions of the equilibrium normalmode displacements, but incorporate anharmonic effects (the magnitudes of the stretch and bend motions are extended until the associated vibrational wavefunctions in these directions provide relative quadrature weights of 1/e). These conditions serve to define the turning-point geometries adopted in the present study. A six-point quadrature employed in the evaluation of the vibrational averaging integral over the vibrational motion provides the desired angle-averaged orbital electron momentum distributions in H20. The calculated vibrational averaging effects on the angle-averaged electron momentum distributions are found to be small, despite the more significant dependence found on molecular geometry shown in fig. 1. This circumstance is largely a consequence of the rapid drop-off in the amplitude of the vibrational wavefunctions away from equilibrium geometry, and of partial cancellations of contributions to the vibrational integral from opposite turning points in the symmetric vibrations. In the harmonic approximation, which is not adopted in the present study, such cancellations would be complete for the symmetric normal-mode vibrations. Small corrections due to vibrational averaging in the momentum distributions of the 3a, and 1bz orbitals in the lowmomentum region shown in fig. 2 appear to reduce the discrepancy between experiment and theory, suggesting that vibrational averaging can have an important, albeit small, effect in this electron momentum region. The small vibrational averaging effects observed in this work are consistent with theoretical studies on vibrational corrections to other electronic properties of I&O, and lend further support to the practice of comparing experimental electron momentum distributions with theoretical values calculated at equilibrium geometries. In view of the noticeable differences between the turning-point and equilibrium momentum distributions of the 3a,, lb? and 2a, orbitals in H20, of the increase in amplitudes at low momentum, in particular, and of the possibility of the non-cancellation of such effects at opposite turning points in other compounds, vibrational averaging should generally be incorporated in refined computational studies of electron momentum distributions in polyatomic molecules. These considerations should also be combined with the more significant effects of finite experimental momentum resolution when comparisons are made with results derived from (e, 2e) spectroscopy measurements.
Acknowledgement Acknowledgement 140
is made to the Natural
Sciences and Engineering
Research Council of Canada, to the Na-
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tional Science Foundation, and to the Department of Energy for partial support of these studies. One of us (KTL) thanks his collaborators for their kind hospitality during his visit to Bloomington, Indiana, in the fall of 1984. The use of facilities of the National Center for Supercomputing Applications is gratefully acknowledged.
References [ 1] [2] [3] [4]
C.E. Brion, Intern. J. Quantum Chem. 29 (1986) 1397. I.E. McCarthy and E. Weigold, Phys. Rept. 27 (1976) 275. I.E. McCarthy and E. Weigold, Contemp. Phys. 244 (1983) 163. K.T. Lcung and C.E. Brian, J. Electron Spectry. 35 ( 1985) 327. [ 51S.Dey, I.E. McCarthy, P-J-0. Teubner and E. Weigold, Phys. Rev. Letters 34 ( 1975) 782. [6] A.O. Bawagan, C.E. Brion, E.R. Davidson and D. Feller, Chem. Phys. 1 I3 (1987) 19. [7] A.O. Bawagan, L.Y. Lee, K.T. Leung and C.E. Brion, Chem. Phys. 99 ( 1985) 367. [ 81J.N. Murrell, S. Carter, SC. Farantos, P. Huxley and A.J.C. Varandas, Molecular potential energy functions (Wiley, New York, 1984) p. 133. [9] D. Feller, CM. Boyle and E.R. Davidson, J. Chem. Phys. 86 (1987) 3424. [IO] C.W. Kern and R.L. Matcha, J. Chem. Phys. 49 (1968) 2081. [1I]W.C.ErmlerandC.W.Kem,J.Chem.Phys.55(1971)4851. [ 121 M. Breitenstein, R.J. Mawhorter, H. Meyer and A. Schweig, Mol. Phys. 57 ( 1986) 8 I. [ 131 P.W. Langhoff, S.R. Langhoff, T.N. Rescigno, J.S. Schirmer, L.S, Cederbaum, W. Domcke and W. von Niessen, Chem. Phys. 58 (1981) 71. [ 141 G.R.J. Williams, I.E. McCarthy and E. Weigold, Chem. Phys. 22 (1977) 281. [ 151I.E. McCarthy, in: Coherence and correlation in atomic collisions, eds. H. Kleinpoppen and J.F. Williams (Plenum Press, New York, 1980) p. 1. [ 161 G.H.F. Diercksen, W.P. Kraemer, T.N. Rescigno, C.F. Bender, B.V. McKay, S.R. Langhoff and P. W. Langhoff, J. Chem. Phys. 76 (1982) 1043. [ 171 K.T. Leung, C.E. Brion, B.W. Fatyga and P.W. Langhoff, Chem. Phys. 96 (I 985) 227.
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