- Email: [email protected]
Electron-correlation effects in electron-scattering cross-section calculations of N2
ELECTRONCORRELATION
EFFECTS
IN ELECTRONSCA-ITERING
CROSSSECTION
CALCULATIONS
Martin BREITENSTEIN, Andreas ENDESFELDER, Amlin SCHWEIG and Werner ZITTLAU Fachbereich Physikdische Received 11 hkuch
27 hlsy 1983
CHEMICAL PHYSICS LEXTERS
Volume 97, number 45
chemie.
Hermann
UnbersirEt Marburg. D-3550
OF N2 +
MEYER,
hfarburg,
West Germany
1983
Total differential hi&-enera electron-scattering cross sections for N2 have been calculated in the first. Born approlimation and compared with available experimental data. The importance of including electron correlation is demonstnted.
and method
of calculation
1. Introduction
2. Basic assumptions
High-energy (40 keV) electron-scattering studies on chemical binding effects still reveal some striking differences between experimental and theoretical results, mainly concerning total (elastic plus inelastic) differential cross sections. The calculations are usually based on molecular Hartree-Fock (I-IF) wavefunctions. esceptions being calculations on Hz [l-3] and CO [4], where electron correlation has been taken into account (a further electron-correlation study on N2 and Hz0 [5] confines itself to elastic scattering). Whereas the calculations on the two-electron system H2 including nearly 100% of the correlation energy may not be regarded as representative for chemical bonds, the multiconfiguration self-consistent field (MC SCF) study on CO [4] representing 2 1% of the correlation energy suffers from the absence of comparable experimental results. Nevertheless, it is generally expected that markedly better agreement between experimental and theoretical results will be attained by using correlated wavefunctions instead of HF wavefunctions. It is the aim of the present work to confirm this expectation using N2 as an example. This molecule has been thoroughly investigated by experimental&s [6,7], and the disagreement between experimental and theoretical HF results [S] is pronounced.
In high-energy electron-scattering physics [9] the relative differential cross section of a molecule in the gas phase, u(s), may be dellned as
*
Part 18 of Densities_
Comparison
of Observed and Calculated
0 009-2614/83/0000-0000/S
Electron
03-00 0 1983 North-Holland
o(s) = I(s)&&)
-
(1)
I(s) and &J(S) are the differential cross sections of the molecule and of a fried electron (i.e. the Rutherford cross section), respectively_ Using atomic units throughout, Id(s) is given as &J(s) = 4/s4 -
(3
The magnitude s of the scattering the scattering angle 0 by s = (47rb) sin@
,
vectors
is related fo
Q)
assuming the electron wavelength h fo be (practically) identical for incident and scattered electrons. Instead of u(s) itself, generally the following difference function Au(s) is of interest in high-precision electron-scattering work: Au(s) = o(s) -
oIAM(s)
-
(4)
represents the relative differential cross section of the independent atom model (IAM) [9] of the molecule considered_ It is expected that for large s values Au(s) vanishes whereas for small s values this function visualizes chemical binding effects. The present paper deals primarily with total (elasqua
403
Volume
97.
CHEBIICAL PHYSICS LETTERS
number 4.5
Iic plus inelastic) electron scattering, corresponding quantities being distinguished by the superscripts tot, cl~sr. and inelast in the formulas. The most reliable experimental Auto*(s) curve of X2 pubhshed so far [7] is related to a uj_&(s) funcIion using atomic partial wave elastic scattering amplirudrs and inelastic scatrering factors in the first Born approuimdtion. both being calculated on the IlF level. Tht hitherto best theoretical .&~~~*(s) curve [S] is based on iAh1 and molecular functions, o&it (s) dnd a”‘(s). calculated in the first Born approximation using atomic HF [lo] and molecular near HF (Slatertype orbirals up 10 f type) [ 111 (for the graphs of both these experimental .md theoretical &‘*(s) curves. see fig. 1. section 3)_ Tilere arc large discrepancies between the experinientJ1 and theoretical Au”‘(s) curves for s values up 10 8 A-1. the maximum deviation occurring in the range from 3 IO 4 a-’ (fig. 1). Besides possible experimental errors (not discussed in the present paper; for 3 critical note. however. see section 3) approximations mdde in the theoretical treatment could be responsible for the poor agreement between theory and experinlent. Concerning the jlrst Born approximation there are ehperunental [6] and theoretical [ 121 indications 0131 rllis approximation holds for tile Au’~*(s) funcnon of Nz within the experimental uncertainties. IV&W~ of nuJeur dwation may be justified. too. since lhe vibratIona effects calculated for the N2 molecule [ 131 and ifs IAM counterpart are very simi1,tr. The most severe approximation that remains to be studied is tlie ncglecr 0-f electrm correlation. For tllese redsons. Cl calculations of Au’“‘(s) for tl~r rigId N2 molecule in the framework of the first Born q’proximation will be presented below.
27 hfay 1983
of nucleus A, respectively. and rl is the position vector of an electron. p (rl) and r(rl, ‘2) denote the oneelectron density (normalized to the total number of electrons, J’V)and the two-electron density [norrnalized to N(N - I)], respectively. In the present work, P(Q) and I’(rl, Q) were calculated on the approshnate Hartree-Fock level (AHF, the near HF calculation mentioned above being a special case of AHF) as well as on the level of subsequent CI calculations. On both levels these densities can be written in the form
(6) and
where oU denotes an orbital of the AHF basis and Pgv aIld %v~O are elements of the one- and two-electron density matrices, respectively, corresponding to the AHF basis. For a molecule in the gas phase utot(s) of eq. (5) must be rotationally averaged. Taking into account eqs. (6) and (7) one obtains P(s) with
= u”“(s) -I-P(s) the pure nuclear
unn(s)=7
the nucleus-electron
J*‘(s)
jo(slrA
.Z_$+ Xp
-
Xp
c - _ _&.l=B
esP[is-h
-‘d
Z_.r
Jp(r*)
exp[is*(rl
-r~)]
drl
Ji
r(r1.n)
Z-4 and r-4
403
exp[u*(rl
-r2)]
are the charge number
drl dr2 +A’_
and position
term
(10)
term
- ~1)
is the zeroth-order spherical Bessel (A/.w) is the one-electron integral
@c(y)= (l/4n)~exp(-is-r~)~~V(s) and (/~lh~>
+
(5)
vector
(9)
(11) function.
$J
term
une(s) = -2 AqV Z~P.JAj.4V) and the pure electronic
;l
w
,
z:+2 Az z_.$zBjO(sjrrf -rBj)
For J molecule with a fixed nuclear geometry and ,! given sc.it teririg vectors the following relationship is v.hd within rhe tirst Born approximation: = c
+ P(s)
the two-electron
dR integral
(12)
Volume 97, number 4.5
CHEMKXL
27 May 1983
PHYSICS LFTKERS
molecule)_ An expression similar to eq. (8) is obtained for oekrst(s):
The atomic elastic and inelastic form factors Fd (s) and 5’~ (s), respectively, were taken from ref. f3-31, thus representing qaasI-exactly the HF level in the fmework of the fit Born approximation_ In order to analyze the Aotot(s) curve the following difference functions are defmed, with the subscripts AEIF and AHF CI referring to the molecular AHF and AHF CI calculations, respectively:
&s*(s)
Au&(s)
with &.&) =s#&l)
#&l)
exp(+~)
d3rl -
(14)
d52 denotes an element of the solid angle corresponding to s (this vector being rotated relative to the fured
= a”(s)
+ u”(s)
+ o”‘(s)
,
cm
With
P’(s)
AQarr(sl= = ,qG
P,~~&iVl?m~ f
(16)
oinelast(s)is simply obtained by taking the difference: &e1=t(S)
= &ot(Sj _ oel=t(Sj
_
(17)
The integmIs of eqs. (12)-(14) were evaluated anaIyticaliy for Cartesian CGTOs (contracted gaussisntype orbitals) of the s, p and d type. Based on these evaluations the FORTRAN program DSIGMA was written which allows for the calculation of Auto*(s), AueIss*(s), and Auinelast (s) (for details see ref. [ 141; a simifar method of integration developed independently from us has been published recently [ 15])_ 45 s values in the range from 0.2 to 20 2i-l were chosen, and the resulting data were piotted using spline interpolation. For the AHF calculations the 4-3 1G [ Ifi] + BF (bond functions) 117,181 basis set was chosen. The subsequent variational CI calculation included all singly and doubly excited configurations. An experimental equilibrium distance of 1.094 a [19] was used. The AHF calculations were performed using the POLYATOM program system [ZO] and the CI calculations using the CI program of refs. [Z 1,22] supplemented by a program for computing the two-electron density matrices (for details see ref. [14])_ The IAM cross section &&(s) was calculated according to the formula [9] &Us)
= &&p(S)
= F
W.. - E:jt (@I2
QAHF cr@)
- u&(s)
(19)
,
- OAIIF~)
f20)
*
In the last equation the superscripts tot. elast, inelasist, ne, and ee, respectively, are to be added. .&J,,,(S) then represents the corresponding correlation effects in the molecule_ According to eqs. (19) and (20) for a molecule treated on the AI-IF CI level, the &sto”(s) function may be writ ten as Aotof(s) = Aoj#P(s)
+ Au:;,(s),
where &r~&(s)
= Au$$(s)
+ &&$st(s)
,
(22)
or alternatively, takinS eq_ @) into account, A&&(S)
= &$&(S)
-t A~=co,r(S) .
(33)
For the discussion of the theoretical and experimental scattering data it is helpful to include Tavards theorem [24], in the form AI’ = .-I
SAdto’
ds ,
(24)
where AV is the difference between the potential energies of the molecule and its IAM counterpart. In the theoretical treatment AV may be partitioned in analogy with A*‘“‘(s) in eq. (2 1): AV= AVAHF + AVmrr ,
cm
where AVAHF refers to the difference of the potential energies of the AHF and the 1A.Mmolecules. and represents the potential energy par* of the A total correlation energy MCOIf_
year,
The43
1G + BF basis set is known to allow for one405
CHEMICAL PHYSICS LETTERS
\‘oIun~e 97. number 4.5 Table 1 Correlation
energies (in au) for Nz
exact (estimated)
-0.538
J-_IlG+BI’Cl
-0.15
[26] 46.7% d,
1;
A r’corr
b,
-1.278 -0.635
=) 2 49.7% =)
=) Tooralelectronic correlarion energy. b, I’otenttal energy parrtof AfTcorr. c) lkriked from the nex IiF results for the total and potential
energies given in ref. [ 271 and the value of tic,, of ref. 176 1. t.Aing the virial theorem into account (which is valid for the exact correlated wavefunction at the experimental equilibrium distance). d) Kcfcrrrd to the exact value of ticorr. ‘.I Kcferred 10 the exact value of AVcorr_
electron
deformation density .4HF calculations of near 1jF quality [ 17,l S] and to yield correlation corrccrions to the one-electron densities of a quality sompsrJble lo that of near HF CI calculations [18]_ Corresponding studies on the two-electron density are m progress [25] _Despite the small size of the 4-3 IG + lW h.~ls set the Cl calculations including all singly .md doubly excited configurations already yield nearly 11.1ifof the estimated exact total correlation energy Ir-.;.W (see table 1). l%us it may be expected that the 1-j 1G + BF CI results on electron scattering obtained III the present work arc reasonable approsimations to she exact results. FIN. 1 displays rhe experimental [7]. the near HF [S ] .md the 4-3 1C + BF &rtot(s) curves (the latter
being based on molecular 4-3 1G + BF wavefunctions and quasi-exact atomic HF wavefunctions for the LAM molecule). Compared to the experimental results the differences between the two theoretical Auto’(s) functions are small except for s values from 6 to 9 A-‘, obviously duz to some deficiencies in the 43 1G -+ BF basis set (these deficiencies
cannot be compensated by calculating the IAM cross section on the same AHF level as the molecular cross section; the
basis set defect on L$&(s)
does not appreciably exthe qualitative agreement between the 4-3 1G + BF and near HF Ac~‘~~(s) curves is quite reasonable in the whole range of s valceed 8 value of O-l!)_
The correlation correction A&$(s) resulting from the 4-3 1G + BF CI calculations is shown in figs. 2a and 2b. The relevant values of this function are uniformly negative and occur in the s range up to 10 .a-‘. i.e. in the same range where the aforementioned theoretical Aotot( s ) curves display their well-pronounced oscillatory behaviour. Note, however, that
w--=-l i
I lg. I_ l.\perimental and theoreticJ aotot(s) 8) c\prrtmcntal: X ne.u tll-: + 4-3 1C + UF.
-106
0
----
curves for Nz.
However,
ues.
i
r-
27 May 1983
--T‘_6-
1
.
.
,
a
10
12
.
1‘
.
.
16
18
,
I%. X4-31G + BF Ci correlation car ections for N2; T * Ao$&(S). (a) + AaE$$s); X Au ErF’(s)@) Y Au&(S); # “$&rW-
Volume 97, number 45
CHEMICAL PHYSICS LETI-ERS
both A&$(s) and Auto*(s) have absolute values of the same order of magnitude. As seen from fig_ 2a the inelastic contribution to A&&(s) predominates at small s values whereas the elastic contribution preponderates at larger s values (the latter being in reasonable agreement with the comparable calculations of ret [5]). Both confributions are of the sanze order of nzagzzitude, izz cozztrast to calczrlatiom on Hz where the effect of electron correlatiolz turns out to be mainly an inelastic one [2] _ Fig_ 2b displays the partitioning of A&$(s) into the one-electron density-dependent term A&&r(s) and the two-electron density-dependent term Au”,,,(s) according to eq. (23). The diagram clearly reflects the important role of correlation effects on the one-electron density, these being even larger than the corresponding two-electron effects. Concerning the partitioning of the calculated total correlation energy the present study yields a ratio of roughly 2 : 1 for the electron-nucleus attraction potential to the electronelectron repulsion potential (visible from fig. 2b, too, via Tavard’s theorem)_ 77rese remits suggest tlzatIikewise the exact total cozifelatiozz energy will be nzore affected b-v an izzcreased electrozz atfractiozz to fite NUclei rather tharz b_v a dinzizzis/zed electron-electron repulsion,
opposite to the usual picture of eiecrrozz
correlatiozz effects. It is of course possible that N2 displays an exceptional correlation behaviour The correlation corrected Auto*(s) function is presented in fig. 3. As AHF basis set defects generally cannot be compensated by correlation effects the 4-3 1G + BF CI correlation correction Au*s&(s) has been added to the near HF AC&&Z(S) function instead of the corresponding 4-3 1G -t BF function_ Further calculations of the Au?&(s) function of N2 using several basis sets of near HF quality [ 14,281 suggest that the near HF curve chosen in this work is nearly identical with the (unknown) exact HF curve_ As seen from fig. 3 tize correlation correctiom leads to an essentiah’y improved agreenzazt betweetz tlzeoq! and experinzent. The first maximum in the near HF curve (at s * 1 A-‘, not existing in the experimental curve), has been removed by electron correlation and the extrema at s = 4 A-’ (minimum) and s = 7 A-’ (maximum) have approached the corresponding experimental data by roughly 40-5010 of the original differences_ At this point, it is noteworthy that the potential
Fig. 3. Experimental and theoretiul btot(~)
27 May 1983
curves for X2_
o experimental; + near HF; x near HF plus 4-3 1G +
BF CI
correlation correction; * near Hf-‘ plus estimated e\act correlation correction. For details, see text.
of the calculated total correlation energy part AV,,,, energy covers just one half of the estimated exact value (see table 1). According to Tavard’s theorem this means that the area corresponding to the exact AU&(s) function must be twice as large as that displayed by the A&&(s) function in figs. 2s and 2b. respectively_ If the assumption is made that the foml of this function is not essentially affected when taking
the remaining electron correlation into account, one simply has to scale this function by a factor of 2 to obtain the exact Aura, function. By adding this correlation correction to the near HF A&‘&(s) function an estimation of the exact theoretical Auto’(s) curve is obtained (see fig_ 3). The agreement of this curve with the experimental one is excellent escept for the region of the first minimum where the rheoretical values still lie somewhat higher than the esperimental ones. Certainly, the assumption made above is a strong one and thus might be responsible for the discrepancies in this s region; however, reaching agreement between the theoretical and experimental curves in the first minimum region would worsen the agreement in other s regions, because of the validity of Tavard’s theorem. In this context a critical note on the experimental results appears to be appropriate- If the first Born approximation is valid in determining Auto*(s) - as inferred from experimental results - then the area under this curve must fulfdl Tavard’s theorem. i.e. must equal the area under the estimated theoretical Auto’(s) 407
CHEMICAL PHYSICS IXI”I-ERS
Volume 97. number 4.5
2urse including this is not rhe
100% efectron
correlation.
Obviously,
case; the potential energy derived from
the eqxrimcnta1 curve is too negative by ~10 eV. This value certainly exceeds the possible integral tnmcation error. as tested by theoretical investigations. Thus we believe that on the experirnentai side there are corrections to be msde. too. Clearly. these corrections will be relatively small compared to the correlariorl correcrioris discussed in the present paper. The results presented in this work need confirmation by using more sophisticated electronic waveiuncrions for N2. In addition. further molecules should be studied to alfo~ for more general statements concerning ~lccrruo-correfatioll effects on molecular electronscatrerrng cross sections. Both sarts of studies are in progress, in rhc preseut authors’ group.
4. Conclusion The total electron-scattering difference function Jo’~‘(s). defined according fo eq. (4). displays, for the Nz molecule. striking discrepancies between espcrimem and theory if tfte calculations are performed on the HF level of approxirttarion. In the present work II KSshown Ihar considerably better agreement is achieved by raking electron correlation into account. Cticulatmns were performed in the first Born approxint.&on with a -1-J 1C + BF Cl wavefunction in&ding .dl singly wd doubly excited ~on~~urations leading
to rtbout one hlf of the exact correlation energy. The eldstic dud inrlasric parts of the correlation correction term to Ao’~*(s ) are of the same order of magnitude. The pJrtltloniug of the correlation correction temt into the one- and two-electron density-dependent pxrs emphasizes rfte spechl importance of the correPatton effect on the one-electron density.. Corresponding& . the ~dculated toEa rorrelarion energy is ntore affected by an tncreased electron attraction to rhe nuclei rstiter than by a dmtntished electron-electron repulaon. in contrast fo rhe usual picture of electroncorrelation effects. Of course. studies on N2 and further molecules xc needed using better wavefunctions in order to corrobordre the results of this paper. A Ju’“‘(s) curve representing the full correlation energy hs been csrimated in a very simple way. This curve agrees astonishingIy well with the experimental one. The remaining discrepa~tc~es ntay be due to smart de& cuxtcics m the e~perintent~1 de~e~linatiort. 40s
27 May 1983
This work was supported
by the Deutsche
For-
schtmgsgemeinschaft (SFB 127) and the Fonds der Chemisehen industrie. The calculations were carried out using the TR 440 computer of the ~echeuzen~rum der UniversitSt Marburg.
References J-W. Liu, J. Chem. P&s. 59 (1973) 1988. J.W. Liu and V.H. Smith Jr., Chem. Phys. Letters45 (1977) 59. W. I;olos. HJ. hfonkhorst and I;. Stiewicz, J. Chem. Ph>s. 77 (1982) 1323s J. Epstein and R.F. Stewart. J- Cbem. Phys. 67 (1977) 4238. F. Hirots, H. Tends and S. ShI&tra, Bulletin of the Faculty of Education, Shizuoka University, Natural Science Series, Vol. 31 (1980) p_ 49. hi. Fink, PG. hloore and D. Gregory, J. Chem. Phys. 71 (1979) 5227, $1. Fink and C. Schmiedekamp. J. Chem. Phys. 71 (1979) 5243. J. Epstein and RI. Stewart. f- Cbem. Phys. 66 (1977) 4057. R.A. Bon&am and hi. Fink. Hi& energy dectron scattering (Van Nostrand, Princeton. 1974). PX Eq+us,T-L_ Gilbert =d C-C-3. Roothun, 3. Cfiem. Phys. 56 (1972) $195. P-E. Cade and A-C. Wabi, AT. Data Nucl. Data Tables 13 (1974) 339_ D.A. I(0111and h1.M.Arvedson. J. Chem. Phys 73 (1980) 3818. J. Epstein and R.F. Stewart. J. Cbem. Phys. 70 (1979) 5515. hl. Breitenstein, A. Endesfelder, H. Meyer, A. Schweig and 1%‘.ZitUau. to be oublisbed. Y, Sasaki, S_ ionaka,~T. IIjIma and hf. Kimura, Intern. J. Quantum Chem. 21 (1982) 475. R. &Wield. W-J. He& ana J.A. Popie, J. Chem. Phys. 54 (1971) 724. H.-L. Waseand A. Schweig. Angew- Cbem. 89 (1977) 264;Angew. Chem. Intern. Ed. 16 (1977) 258. hi. Breitenstein, H. Danni5h1,ff, Meyer, A. Schweig, R. Seeger, U_ Seeger and W. Zittlau. to be published. G. Herzberg, Diatomic molecules (Van Nostrand, Princeton. 1950). 1-G. CsizmadIa, M.C. Harrison, J-W. hloskowitz and B.T. SutcIIffe, Theoret. Chim. Acta 6 (1966) 191. H--L. Hase, G. Lnuer. K-W. Scbulte and A. Schweig, Theoret. Chim. Acta 48 (1978) 47. G. Lauer, P_-IV.Schulte and A. Scbweig. J. Am, Chem. Sot. 100 (1978) 4925.
Volume 97. number 4J
CHEhiICAL
[231 C_ Tavard. D. Nicolas and hf. Rouault, J. Chim. Phys 64 (1967) 540. 124) C. Tavard, Cab. Phys 20 (1965) 397. [25] M. Breitenstein, H. Meyer and A. Schweig, to be published.
PHYSICS LETTERS
27 May 1983
1261 S. Wilson and DM_ Silver, J. Chem. Phys 67 (1977) 1689. 1271 PE. Cade, K-D. S&s and A-C. W&l. J. Chem. Phys_ 44 (1966) 1973. 1281 P. Pulay, R. Mawhorter, DA. Kohl and hi. Fink. J. Chcm. PhYs, to be published.
EFFECTS
IN ELECTRONSCA-ITERING
CROSSSECTION
CALCULATIONS
Martin BREITENSTEIN, Andreas ENDESFELDER, Amlin SCHWEIG and Werner ZITTLAU Fachbereich Physikdische Received 11 hkuch
27 hlsy 1983
CHEMICAL PHYSICS LEXTERS
Volume 97, number 45
chemie.
Hermann
UnbersirEt Marburg. D-3550
OF N2 +
MEYER,
hfarburg,
West Germany
1983
Total differential hi&-enera electron-scattering cross sections for N2 have been calculated in the first. Born approlimation and compared with available experimental data. The importance of including electron correlation is demonstnted.
and method
of calculation
1. Introduction
2. Basic assumptions
High-energy (40 keV) electron-scattering studies on chemical binding effects still reveal some striking differences between experimental and theoretical results, mainly concerning total (elastic plus inelastic) differential cross sections. The calculations are usually based on molecular Hartree-Fock (I-IF) wavefunctions. esceptions being calculations on Hz [l-3] and CO [4], where electron correlation has been taken into account (a further electron-correlation study on N2 and Hz0 [5] confines itself to elastic scattering). Whereas the calculations on the two-electron system H2 including nearly 100% of the correlation energy may not be regarded as representative for chemical bonds, the multiconfiguration self-consistent field (MC SCF) study on CO [4] representing 2 1% of the correlation energy suffers from the absence of comparable experimental results. Nevertheless, it is generally expected that markedly better agreement between experimental and theoretical results will be attained by using correlated wavefunctions instead of HF wavefunctions. It is the aim of the present work to confirm this expectation using N2 as an example. This molecule has been thoroughly investigated by experimental&s [6,7], and the disagreement between experimental and theoretical HF results [S] is pronounced.
In high-energy electron-scattering physics [9] the relative differential cross section of a molecule in the gas phase, u(s), may be dellned as
*
Part 18 of Densities_
Comparison
of Observed and Calculated
0 009-2614/83/0000-0000/S
Electron
03-00 0 1983 North-Holland
o(s) = I(s)&&)
-
(1)
I(s) and &J(S) are the differential cross sections of the molecule and of a fried electron (i.e. the Rutherford cross section), respectively_ Using atomic units throughout, Id(s) is given as &J(s) = 4/s4 -
(3
The magnitude s of the scattering the scattering angle 0 by s = (47rb) sin@
,
vectors
is related fo
Q)
assuming the electron wavelength h fo be (practically) identical for incident and scattered electrons. Instead of u(s) itself, generally the following difference function Au(s) is of interest in high-precision electron-scattering work: Au(s) = o(s) -
oIAM(s)
-
(4)
represents the relative differential cross section of the independent atom model (IAM) [9] of the molecule considered_ It is expected that for large s values Au(s) vanishes whereas for small s values this function visualizes chemical binding effects. The present paper deals primarily with total (elasqua
403
Volume
97.
CHEBIICAL PHYSICS LETTERS
number 4.5
Iic plus inelastic) electron scattering, corresponding quantities being distinguished by the superscripts tot, cl~sr. and inelast in the formulas. The most reliable experimental Auto*(s) curve of X2 pubhshed so far [7] is related to a uj_&(s) funcIion using atomic partial wave elastic scattering amplirudrs and inelastic scatrering factors in the first Born approuimdtion. both being calculated on the IlF level. Tht hitherto best theoretical .&~~~*(s) curve [S] is based on iAh1 and molecular functions, o&it (s) dnd a”‘(s). calculated in the first Born approximation using atomic HF [lo] and molecular near HF (Slatertype orbirals up 10 f type) [ 111 (for the graphs of both these experimental .md theoretical &‘*(s) curves. see fig. 1. section 3)_ Tilere arc large discrepancies between the experinientJ1 and theoretical Au”‘(s) curves for s values up 10 8 A-1. the maximum deviation occurring in the range from 3 IO 4 a-’ (fig. 1). Besides possible experimental errors (not discussed in the present paper; for 3 critical note. however. see section 3) approximations mdde in the theoretical treatment could be responsible for the poor agreement between theory and experinlent. Concerning the jlrst Born approximation there are ehperunental [6] and theoretical [ 121 indications 0131 rllis approximation holds for tile Au’~*(s) funcnon of Nz within the experimental uncertainties. IV&W~ of nuJeur dwation may be justified. too. since lhe vibratIona effects calculated for the N2 molecule [ 131 and ifs IAM counterpart are very simi1,tr. The most severe approximation that remains to be studied is tlie ncglecr 0-f electrm correlation. For tllese redsons. Cl calculations of Au’“‘(s) for tl~r rigId N2 molecule in the framework of the first Born q’proximation will be presented below.
27 hfay 1983
of nucleus A, respectively. and rl is the position vector of an electron. p (rl) and r(rl, ‘2) denote the oneelectron density (normalized to the total number of electrons, J’V)and the two-electron density [norrnalized to N(N - I)], respectively. In the present work, P(Q) and I’(rl, Q) were calculated on the approshnate Hartree-Fock level (AHF, the near HF calculation mentioned above being a special case of AHF) as well as on the level of subsequent CI calculations. On both levels these densities can be written in the form
(6) and
where oU denotes an orbital of the AHF basis and Pgv aIld %v~O are elements of the one- and two-electron density matrices, respectively, corresponding to the AHF basis. For a molecule in the gas phase utot(s) of eq. (5) must be rotationally averaged. Taking into account eqs. (6) and (7) one obtains P(s) with
= u”“(s) -I-P(s) the pure nuclear
unn(s)=7
the nucleus-electron
J*‘(s)
jo(slrA
.Z_$+ Xp
-
Xp
c - _ _&.l=B
esP[is-h
-‘d
Z_.r
Jp(r*)
exp[is*(rl
-r~)]
drl
Ji
r(r1.n)
Z-4 and r-4
403
exp[u*(rl
-r2)]
are the charge number
drl dr2 +A’_
and position
term
(10)
term
- ~1)
is the zeroth-order spherical Bessel (A/.w) is the one-electron integral
@c(y)= (l/4n)~exp(-is-r~)~~V(s) and (/~lh~>
+
(5)
vector
(9)
(11) function.
$J
term
une(s) = -2 AqV Z~P.JAj.4V) and the pure electronic
;l
w
,
z:+2 Az z_.$zBjO(sjrrf -rBj)
For J molecule with a fixed nuclear geometry and ,! given sc.it teririg vectors the following relationship is v.hd within rhe tirst Born approximation: = c
+ P(s)
the two-electron
dR integral
(12)
Volume 97, number 4.5
CHEMKXL
27 May 1983
PHYSICS LFTKERS
molecule)_ An expression similar to eq. (8) is obtained for oekrst(s):
The atomic elastic and inelastic form factors Fd (s) and 5’~ (s), respectively, were taken from ref. f3-31, thus representing qaasI-exactly the HF level in the fmework of the fit Born approximation_ In order to analyze the Aotot(s) curve the following difference functions are defmed, with the subscripts AEIF and AHF CI referring to the molecular AHF and AHF CI calculations, respectively:
&s*(s)
Au&(s)
with &.&) =s#&l)
#&l)
exp(+~)
d3rl -
(14)
d52 denotes an element of the solid angle corresponding to s (this vector being rotated relative to the fured
= a”(s)
+ u”(s)
+ o”‘(s)
,
cm
With
P’(s)
AQarr(sl= = ,qG
P,~~&iVl?m~ f
(16)
oinelast(s)is simply obtained by taking the difference: &e1=t(S)
= &ot(Sj _ oel=t(Sj
_
(17)
The integmIs of eqs. (12)-(14) were evaluated anaIyticaliy for Cartesian CGTOs (contracted gaussisntype orbitals) of the s, p and d type. Based on these evaluations the FORTRAN program DSIGMA was written which allows for the calculation of Auto*(s), AueIss*(s), and Auinelast (s) (for details see ref. [ 141; a simifar method of integration developed independently from us has been published recently [ 15])_ 45 s values in the range from 0.2 to 20 2i-l were chosen, and the resulting data were piotted using spline interpolation. For the AHF calculations the 4-3 1G [ Ifi] + BF (bond functions) 117,181 basis set was chosen. The subsequent variational CI calculation included all singly and doubly excited configurations. An experimental equilibrium distance of 1.094 a [19] was used. The AHF calculations were performed using the POLYATOM program system [ZO] and the CI calculations using the CI program of refs. [Z 1,22] supplemented by a program for computing the two-electron density matrices (for details see ref. [14])_ The IAM cross section &&(s) was calculated according to the formula [9] &Us)
= &&p(S)
= F
W.. - E:jt (@I2
QAHF cr@)
- u&(s)
(19)
,
- OAIIF~)
f20)
*
In the last equation the superscripts tot. elast, inelasist, ne, and ee, respectively, are to be added. .&J,,,(S) then represents the corresponding correlation effects in the molecule_ According to eqs. (19) and (20) for a molecule treated on the AI-IF CI level, the &sto”(s) function may be writ ten as Aotof(s) = Aoj#P(s)
+ Au:;,(s),
where &r~&(s)
= Au$$(s)
+ &&$st(s)
,
(22)
or alternatively, takinS eq_ @) into account, A&&(S)
= &$&(S)
-t A~=co,r(S) .
(33)
For the discussion of the theoretical and experimental scattering data it is helpful to include Tavards theorem [24], in the form AI’ = .-I
SAdto’
ds ,
(24)
where AV is the difference between the potential energies of the molecule and its IAM counterpart. In the theoretical treatment AV may be partitioned in analogy with A*‘“‘(s) in eq. (2 1): AV= AVAHF + AVmrr ,
cm
where AVAHF refers to the difference of the potential energies of the AHF and the 1A.Mmolecules. and represents the potential energy par* of the A total correlation energy MCOIf_
year,
The43
1G + BF basis set is known to allow for one405
CHEMICAL PHYSICS LETTERS
\‘oIun~e 97. number 4.5 Table 1 Correlation
energies (in au) for Nz
exact (estimated)
-0.538
J-_IlG+BI’Cl
-0.15
[26] 46.7% d,
1;
A r’corr
b,
-1.278 -0.635
=) 2 49.7% =)
=) Tooralelectronic correlarion energy. b, I’otenttal energy parrtof AfTcorr. c) lkriked from the nex IiF results for the total and potential
energies given in ref. [ 271 and the value of tic,, of ref. 176 1. t.Aing the virial theorem into account (which is valid for the exact correlated wavefunction at the experimental equilibrium distance). d) Kcfcrrrd to the exact value of ticorr. ‘.I Kcferred 10 the exact value of AVcorr_
electron
deformation density .4HF calculations of near 1jF quality [ 17,l S] and to yield correlation corrccrions to the one-electron densities of a quality sompsrJble lo that of near HF CI calculations [18]_ Corresponding studies on the two-electron density are m progress [25] _Despite the small size of the 4-3 IG + lW h.~ls set the Cl calculations including all singly .md doubly excited configurations already yield nearly 11.1ifof the estimated exact total correlation energy Ir-.;.W (see table 1). l%us it may be expected that the 1-j 1G + BF CI results on electron scattering obtained III the present work arc reasonable approsimations to she exact results. FIN. 1 displays rhe experimental [7]. the near HF [S ] .md the 4-3 1C + BF &rtot(s) curves (the latter
being based on molecular 4-3 1G + BF wavefunctions and quasi-exact atomic HF wavefunctions for the LAM molecule). Compared to the experimental results the differences between the two theoretical Auto’(s) functions are small except for s values from 6 to 9 A-‘, obviously duz to some deficiencies in the 43 1G -+ BF basis set (these deficiencies
cannot be compensated by calculating the IAM cross section on the same AHF level as the molecular cross section; the
basis set defect on L$&(s)
does not appreciably exthe qualitative agreement between the 4-3 1G + BF and near HF Ac~‘~~(s) curves is quite reasonable in the whole range of s valceed 8 value of O-l!)_
The correlation correction A&$(s) resulting from the 4-3 1G + BF CI calculations is shown in figs. 2a and 2b. The relevant values of this function are uniformly negative and occur in the s range up to 10 .a-‘. i.e. in the same range where the aforementioned theoretical Aotot( s ) curves display their well-pronounced oscillatory behaviour. Note, however, that
w--=-l i
I lg. I_ l.\perimental and theoreticJ aotot(s) 8) c\prrtmcntal: X ne.u tll-: + 4-3 1C + UF.
-106
0
----
curves for Nz.
However,
ues.
i
r-
27 May 1983
--T‘_6-
1
.
.
,
a
10
12
.
1‘
.
.
16
18
,
I%. X4-31G + BF Ci correlation car ections for N2; T * Ao$&(S). (a) + AaE$$s); X Au ErF’(s)@) Y Au&(S); # “$&rW-
Volume 97, number 45
CHEMICAL PHYSICS LETI-ERS
both A&$(s) and Auto*(s) have absolute values of the same order of magnitude. As seen from fig_ 2a the inelastic contribution to A&&(s) predominates at small s values whereas the elastic contribution preponderates at larger s values (the latter being in reasonable agreement with the comparable calculations of ret [5]). Both confributions are of the sanze order of nzagzzitude, izz cozztrast to calczrlatiom on Hz where the effect of electron correlatiolz turns out to be mainly an inelastic one [2] _ Fig_ 2b displays the partitioning of A&$(s) into the one-electron density-dependent term A&&r(s) and the two-electron density-dependent term Au”,,,(s) according to eq. (23). The diagram clearly reflects the important role of correlation effects on the one-electron density, these being even larger than the corresponding two-electron effects. Concerning the partitioning of the calculated total correlation energy the present study yields a ratio of roughly 2 : 1 for the electron-nucleus attraction potential to the electronelectron repulsion potential (visible from fig. 2b, too, via Tavard’s theorem)_ 77rese remits suggest tlzatIikewise the exact total cozifelatiozz energy will be nzore affected b-v an izzcreased electrozz atfractiozz to fite NUclei rather tharz b_v a dinzizzis/zed electron-electron repulsion,
opposite to the usual picture of eiecrrozz
correlatiozz effects. It is of course possible that N2 displays an exceptional correlation behaviour The correlation corrected Auto*(s) function is presented in fig. 3. As AHF basis set defects generally cannot be compensated by correlation effects the 4-3 1G + BF CI correlation correction Au*s&(s) has been added to the near HF AC&&Z(S) function instead of the corresponding 4-3 1G -t BF function_ Further calculations of the Au?&(s) function of N2 using several basis sets of near HF quality [ 14,281 suggest that the near HF curve chosen in this work is nearly identical with the (unknown) exact HF curve_ As seen from fig. 3 tize correlation correctiom leads to an essentiah’y improved agreenzazt betweetz tlzeoq! and experinzent. The first maximum in the near HF curve (at s * 1 A-‘, not existing in the experimental curve), has been removed by electron correlation and the extrema at s = 4 A-’ (minimum) and s = 7 A-’ (maximum) have approached the corresponding experimental data by roughly 40-5010 of the original differences_ At this point, it is noteworthy that the potential
Fig. 3. Experimental and theoretiul btot(~)
27 May 1983
curves for X2_
o experimental; + near HF; x near HF plus 4-3 1G +
BF CI
correlation correction; * near Hf-‘ plus estimated e\act correlation correction. For details, see text.
of the calculated total correlation energy part AV,,,, energy covers just one half of the estimated exact value (see table 1). According to Tavard’s theorem this means that the area corresponding to the exact AU&(s) function must be twice as large as that displayed by the A&&(s) function in figs. 2s and 2b. respectively_ If the assumption is made that the foml of this function is not essentially affected when taking
the remaining electron correlation into account, one simply has to scale this function by a factor of 2 to obtain the exact Aura, function. By adding this correlation correction to the near HF A&‘&(s) function an estimation of the exact theoretical Auto’(s) curve is obtained (see fig_ 3). The agreement of this curve with the experimental one is excellent escept for the region of the first minimum where the rheoretical values still lie somewhat higher than the esperimental ones. Certainly, the assumption made above is a strong one and thus might be responsible for the discrepancies in this s region; however, reaching agreement between the theoretical and experimental curves in the first minimum region would worsen the agreement in other s regions, because of the validity of Tavard’s theorem. In this context a critical note on the experimental results appears to be appropriate- If the first Born approximation is valid in determining Auto*(s) - as inferred from experimental results - then the area under this curve must fulfdl Tavard’s theorem. i.e. must equal the area under the estimated theoretical Auto’(s) 407
CHEMICAL PHYSICS IXI”I-ERS
Volume 97. number 4.5
2urse including this is not rhe
100% efectron
correlation.
Obviously,
case; the potential energy derived from
the eqxrimcnta1 curve is too negative by ~10 eV. This value certainly exceeds the possible integral tnmcation error. as tested by theoretical investigations. Thus we believe that on the experirnentai side there are corrections to be msde. too. Clearly. these corrections will be relatively small compared to the correlariorl correcrioris discussed in the present paper. The results presented in this work need confirmation by using more sophisticated electronic waveiuncrions for N2. In addition. further molecules should be studied to alfo~ for more general statements concerning ~lccrruo-correfatioll effects on molecular electronscatrerrng cross sections. Both sarts of studies are in progress, in rhc preseut authors’ group.
4. Conclusion The total electron-scattering difference function Jo’~‘(s). defined according fo eq. (4). displays, for the Nz molecule. striking discrepancies between espcrimem and theory if tfte calculations are performed on the HF level of approxirttarion. In the present work II KSshown Ihar considerably better agreement is achieved by raking electron correlation into account. Cticulatmns were performed in the first Born approxint.&on with a -1-J 1C + BF Cl wavefunction in&ding .dl singly wd doubly excited ~on~~urations leading
to rtbout one hlf of the exact correlation energy. The eldstic dud inrlasric parts of the correlation correction term to Ao’~*(s ) are of the same order of magnitude. The pJrtltloniug of the correlation correction temt into the one- and two-electron density-dependent pxrs emphasizes rfte spechl importance of the correPatton effect on the one-electron density.. Corresponding& . the ~dculated toEa rorrelarion energy is ntore affected by an tncreased electron attraction to rhe nuclei rstiter than by a dmtntished electron-electron repulaon. in contrast fo rhe usual picture of electroncorrelation effects. Of course. studies on N2 and further molecules xc needed using better wavefunctions in order to corrobordre the results of this paper. A Ju’“‘(s) curve representing the full correlation energy hs been csrimated in a very simple way. This curve agrees astonishingIy well with the experimental one. The remaining discrepa~tc~es ntay be due to smart de& cuxtcics m the e~perintent~1 de~e~linatiort. 40s
27 May 1983
This work was supported
by the Deutsche
For-
schtmgsgemeinschaft (SFB 127) and the Fonds der Chemisehen industrie. The calculations were carried out using the TR 440 computer of the ~echeuzen~rum der UniversitSt Marburg.
References J-W. Liu, J. Chem. P&s. 59 (1973) 1988. J.W. Liu and V.H. Smith Jr., Chem. Phys. Letters45 (1977) 59. W. I;olos. HJ. hfonkhorst and I;. Stiewicz, J. Chem. Ph>s. 77 (1982) 1323s J. Epstein and R.F. Stewart. J- Cbem. Phys. 67 (1977) 4238. F. Hirots, H. Tends and S. ShI&tra, Bulletin of the Faculty of Education, Shizuoka University, Natural Science Series, Vol. 31 (1980) p_ 49. hi. Fink, PG. hloore and D. Gregory, J. Chem. Phys. 71 (1979) 5227, $1. Fink and C. Schmiedekamp. J. Chem. Phys. 71 (1979) 5243. J. Epstein and RI. Stewart. f- Cbem. Phys. 66 (1977) 4057. R.A. Bon&am and hi. Fink. Hi& energy dectron scattering (Van Nostrand, Princeton. 1974). PX Eq+us,T-L_ Gilbert =d C-C-3. Roothun, 3. Cfiem. Phys. 56 (1972) $195. P-E. Cade and A-C. Wabi, AT. Data Nucl. Data Tables 13 (1974) 339_ D.A. I(0111and h1.M.Arvedson. J. Chem. Phys 73 (1980) 3818. J. Epstein and R.F. Stewart. J. Cbem. Phys. 70 (1979) 5515. hl. Breitenstein, A. Endesfelder, H. Meyer, A. Schweig and 1%‘.ZitUau. to be oublisbed. Y, Sasaki, S_ ionaka,~T. IIjIma and hf. Kimura, Intern. J. Quantum Chem. 21 (1982) 475. R. &Wield. W-J. He& ana J.A. Popie, J. Chem. Phys. 54 (1971) 724. H.-L. Waseand A. Schweig. Angew- Cbem. 89 (1977) 264;Angew. Chem. Intern. Ed. 16 (1977) 258. hi. Breitenstein, H. Danni5h1,ff, Meyer, A. Schweig, R. Seeger, U_ Seeger and W. Zittlau. to be published. G. Herzberg, Diatomic molecules (Van Nostrand, Princeton. 1950). 1-G. CsizmadIa, M.C. Harrison, J-W. hloskowitz and B.T. SutcIIffe, Theoret. Chim. Acta 6 (1966) 191. H--L. Hase, G. Lnuer. K-W. Scbulte and A. Schweig, Theoret. Chim. Acta 48 (1978) 47. G. Lauer, P_-IV.Schulte and A. Scbweig. J. Am, Chem. Sot. 100 (1978) 4925.
Volume 97. number 4J
CHEhiICAL
[231 C_ Tavard. D. Nicolas and hf. Rouault, J. Chim. Phys 64 (1967) 540. 124) C. Tavard, Cab. Phys 20 (1965) 397. [25] M. Breitenstein, H. Meyer and A. Schweig, to be published.
PHYSICS LETTERS
27 May 1983
1261 S. Wilson and DM_ Silver, J. Chem. Phys 67 (1977) 1689. 1271 PE. Cade, K-D. S&s and A-C. W&l. J. Chem. Phys_ 44 (1966) 1973. 1281 P. Pulay, R. Mawhorter, DA. Kohl and hi. Fink. J. Chcm. PhYs, to be published.