Radial analysis of the electric field gradient at the Cl nucleus in HCl

Volume 37, number 3

RADlAL

CHEMICALPHYSICS LETTERS

ANALYSIS OF THE ELECTRIC FIELD GRADlENT

J.E. GRABENSTETT’ER* Chemisrry Dcparmcnr,

1 February

1976

AT THE Cl NUCLEUS LN HCl

and MA. WHiTEHEAD

McGUl

Vnlniversiry. Irfoonrreal, Quebec. Camda,

H3C 3GI

Received 26 August 1975

An OCE SCF calculation is shown to arise principally in HD.

on HCI has been performed using P 29-ST0 basis. The electric field gradient at tic Cl nucleus from electron density very close to the Cl nucleus, as in ND’, but in contrrrst to the situation

1. introduction

evaluating the contributions density

Two opposing tendencies determine the electronic expectation value of the electric field gradient (EFG) at a nucleus. The operator for this property, 4 = (%os% - I)/%3 ,

0)

with r measured from the nucleus and 0 from the principal axis of the EFG tensor, is large for small r because of the r- 3 factor. This suaesls that the expectation value of (1) over an electronic ~vavefunction (qeIec) is due mainfy to electron density very close to the nucleus. However, (1)

can be rewritten as

4 = 4 (iT,!5)“*szo (&#)r-3 ,

(2)

where Szo is the real spherical harmonic with i = 2, and r?t = 0. Since real spherica harmonics are orthonormal, the expectation value of (2) for a spherically symmetric electron distribution (with angular dependence SOO) is zero. This suggests that the electron density ciose to a nucleus such as Cl, consisting mostly Gf nearly spherically symmetric inner electron Shells, contributes IiftIe to qcIs, and that qel, is mainly due to electron density in the bonding region, where deviations from spherical symmetry. are large. Which effect predominates can be determined by = Present address: Chemistry Department, University of New Brunswick, Fredericton; New Brunswick, Canada. E3B 5A3.

contained

to qeIcT from the electron

in a series of nested

splzerical

shlls

centred at the nucleus. Such an analysis has not previously been performed for a molecule containing Cl; the only quadrupoiar nucleus for which radial analysis of qdec has been performed is the deuteron in Hi)+ and HD [I ]. The present paper reports a radial analysis of qelec at Cl for a one-center expansion SCF (OCE SCF) wavefunction for HCl.

2. The

wavefunction

The wavefunction, reported in more detail elsewhere [2], was calculated at the equilibrium internuclear distance of 2.4087 bohr [3,4] ; the basis set consisted of 29 Slater-type orbitals (STCYs)centred on the Cl nucleus (19 u orbitals, 5 z pairs). 20 of these (IO ET orbitals, 5 pi pairs) formed the basis optimized by Gilbert and Wahl for Cl, [S] ; this basis was augmented by adding u orbit& with principal quultum-number IZ= 9, alf allowed 1 values, and the orbital exponent chosen so -&at the radial factor has a m~mum at the internuclear distance to allow ffie wavefunction to peak as much as possible st the proton. The SCF ener,ey in this basis is -460.06894 hartree. The calculated qelec from this wa+zfunction is 3.494 2u, compared to the experimental value of 3-491 au (these quantities are both electronic expectation values; the point-charge &clear contribution has been sub-

1 February

CHE?JICAL PHYSICS LETTERS

Volume 37. number 3

1976

.tr&ted

ciently fine analysis of Qei~ without causing numeri-

ment is fortuitous,

cal error in evaluating the integrals in (7). C,z is the contribution to qelK from electron density in the hoi!UW sphericul shell centred at the EFG nuclues and satisfying fn-1)h 0. If xi and xi are both Is orbit&, a k = -1 integral formally arises but in this case the angular factor vanishes. These integrals are easily evaluated, although care is needed to avoid numerical errors due to can~e~ation. This is mt a nunle~~~l integration method for qelL., since ezch C,, is evaluated analytically; the purpose in writing (7) is to obtain the

from the experimental v&e). Such close agreebut it indicates that the wavefunc‘, tion yields an accurate enough qeL_ to make radial analysis worthwh.iIe. The near Hartree-Fock HCl wavefunctiort of McLean and Yorhimine gives qelec = 3.419 au [6].

.3. Radial analysis of qek For a &ngIe-determinant wavefu~ction,

molecular orbital, MO,

(3)

contributions with k- ranging over the occupied MO’s and with Nk the occupation number of MO Qk. In an OCE basis the $JX_are linear combinations of uz basis orbitals Xi of the form Xi =& (I?) ‘lj,,‘i

(’ 39) I

(4)

with r, 0, 0, and S I,,~ as in (1) and (2) if the xi are cen.atred on the EEG.nucicus; Ok is expressed as

to qelec from individual spherical shells.

4. Results The interval size 12in (7) was chosen as 0.1 bohr; the first five C,, and the corresponding cumulative

EFG’s 48)

and (3) becomes, using (2) and (411and orating tegrals in (3) explicitly,

the in-

are given in tabie 1. The magnitude of all Ctl with II > 5 is less than 0.0 15 au; the C,z are negative for /I = 6 and ?I > 16. About 96% of qclec comes from the

sphere r < 0.3 bohr. The function CA (r) defined in (6) has a rn~nlu~n at 0.06 bohr and drops rapidly; it is piotted in fig. 1. due to electron Thus for HCI, qelec is principally density extremely close to the Cl nucleus. The moSince the ‘bracketed term is a funcrian can be rewritten as

cf r alone,

(5)

D11

Q&c = _’

sdrC,W,

(6)

Tziblc 1 fntc_crals o-ax intervals in r of charge density weigbttcd by 6. Qunntirics

defied

in eqs. (7) and (8) of text

0

which &n be expces%zd as.a sum oi’co~tr~butions finite intervals in r -=

from

nh

9clrr=C f rt=.t (,I_-f)h

drC,

@)= 2 C,, , n=‘l

(7).

‘where h ,is an interval size chosen to prbvide a suffi:_. 1 548.:

lecular electron densi Ly can be regarded as arising from three electron shells, (i) consisting of the 1 u MO (es-

_,..‘i.’ ,-’ .,_ ,‘_

.., : :

‘.

.‘,

,f

r(bohr) 1 2

0.0-0.1 0.1 - 0.2

3

0.2-0.3

4 5 >5

0.3-0.4 0.4-0.5

1.6976 1 1.23684 0.41510 0.09868 0.0~145 <0.015

1.69761 2.93446 3.34955 3.44823 3.45968

--we-

:

Volume

37. number

3

CfIE!.ZICAL

PHYSICS

LETTERS symmetry

by the proton.

The CR andg,

of table

1

anaiysed by shefl in table 2; 8470 of that part of qclec due to e!ectron density within 0.3 bohr of the nucleus comes from the third (va!ence) electron shell. are

In table 3 the efecrron

density is analysed

by hell,

with pI1 co~es?onding to qll in (8) when 4 is replaced by the identity operator in (3); orzlf’ 3% of^che efecrrorr density withirr 0.3 bohr of rhe CI nucleus comes from rhe rhird elecrron shell. Thus although qelec in HCl comes principally from electron density near the Cl nucleus, indicating dominance of the radial factor in (I), the angular factor reduces the ~ont~butions of the nearly spherically symmetric inner she!k, and qcIec is mainly due to the small fraction of the eiec’ tron density in valence MO’s but near the Cl nucleus.

5. Discussion Fig. 1. Charge density weighted by 4; angukuvaii~b!es grated Out. JF dr CA (f) = 4elec.

inte-

scntially a filled Cl- K shell); {ii} the 20, 3a, and l?i ?JO’s (essentialijr the filled CI- L shell); and (iii) the 40, 5u, and 2~ kfO’s, which correspond to the filled M shell of Cl-, distorted significantly from spherical Table 2 P~tition~g

of the

tZ

1 3 4 5

qn of table I into contributions from efcotron skefls ._-r(bohr) 1st shell 4n (=I _______

2

-_.-

-.-__.

0.0-0.1 0.1-0.2 O-2-0.3 0.3 - 0.4 0.4 - 0.5 ~_-._--_---

Table 3 Shell analysis

of the unweigilted

ti

r(bohr)

1 2 3 4 5

Several authors [I ,8] have shown that qeIec at the deuteron.in HD is principally due to electron density midway between tGe nuclei rather than very close to the deuteron. Henderson and Ebbing [lf have presented a graph simiiar to fig. 1 for qctcc at D from an OCE wavefunction for HD at R = I A bohr in a basis of ST05 with non-integral principa! quantum num

o.o-0.l 0.1-0.2 0.2-O-3 0.3--0.4 014 - 0.5

elecrron

_--------..

-0.00003

1.69761 2.93446 3.34955 3.44823 3.45968

--

---

-0.00004 -0.00005 -0.00005 -0.00005

2nd ~fiell

3rd shell ---___-

0.29437

1.4@3’-6

0.48155 0.53496 OS4499 0.54454 ----~---

2.45295 2.81464 2.90329 2.91519

density a) Pn !=) -

1st sbfiel: b)

1.43298 2.84220 4.60074 6.47750 7.96303

1.27911 1.91446 1.99260 1.99942 1.99995

0a’(n) dr ig sine d@ fg” “)P(F)=Xki*‘k9~{~);Pn=lo b) 1st shelf occupied by two electrons c) 2nd and 3rd shells occupied by eight electrons each.

2nd G-K% =)

3rd sheil c) -

d#p(F).

0.14333 0.87400 2.46177, 4.26940 5.73280

0.01054 0.05374 0.14038 0.20868 0.23028

Volume 37, number

I

3

CHEMICAL

PHYSiCS LETTERS

b&s. They [I] found that CA (r) peaked at 0.8 bohr from the deuteron (57% of the in temuclear distance), in.contrast to the 2.5% found above for HCI. How.’ ever, a similar dculation of HDf ,which, like HCI, ‘can be regarded as arr atom (or ion) distorted by a bare proton, gave C,(r) peaking at 20% of the internuclear distance [l], farther than our finding for HCl but closer than for HD. Thus for molecules such a. HD, which can be regarded as sphericahy symn LCricelectrr,n distributions about quadrupolar nuclei distorted by’ the approach of a neutral atom, the valence MO’s are not strongly distorted from spherical symmetry close to the quadrupoIar nucleus, and qrtec is mainly due to electron density in the bonding region. In molecules such as HD+ and HCl, which can be regarded as spherical electron distributions abcut quadrupolar nuclei distorted by the approach of ;1positive ion, the valence MO’s are strongly distorted close to the quadrupolar nucleus [9], and qek_ is due mainly to electron density close to the quad~p~~lar nucleus. Consequently one expects an accurate value for nelec to be obtained with a wavefunction which is not particularly accurate close to the quadrupolar nucleus, for molecules

such OS I-D

[IO],

but not for molecules

such as HD’ or HCI. The excellent SCF value for qelLx at CI in HCI (3.35 au) obtained by Rothenberg et al. [l l] in a two-centre basis of Cartesian gaussian functions indicates that suGh a basis, although lacking cusps at the nuclei, cm :epresent the electron density reasonably weif near nuclei. Simifar remarks apply to Bishop’s rather accurate calculation of qcIec in HDf using car-

.r

1 February

1916

tesian gaussian functions centred symmetrically about the bond centre a short distance from the nuclei [S]. The poor qelec at Cl in HCl obtained by Petke and Written using gaussian lobe functions to approximate p and d orbitals [I21 sugests that these functions do lrot represent ;he electron density well near nuclei.

Helpful discussions with Professor V.H. Smith Jr. are gratefully acknowledged. This research was supported by the National Research Council of Canada.

References [ 11 R.C. llrnderson and D.D. Ebbing, J. Chcm. Phys. 47 f 1967) 69. [Z] J.E. Crzbensterter and hf.\. Whitehead, to be published. [3] E.W. Kaiser, J. Chem. Phys. 53 (1970) 1686. ]4] P.R. Bunker, J. Mol. Spectry. 39 (1971) 90. [S] T.L. Gilbert and AC. Wahl, J. Chem. Phys. 55 (19.71) 5217. [6] A.D. hlclean and %f. Yos!!iminc, J. Chem. Phys. 47 (1967) 3256. [7] J..4. Gaunt, Phil. Trans. Roy. Sot. 2281% (1929) 151. [B] UN. Bitiop, J. Chem. Phys. 49 (1968) 3718. [9] M.A. Whitchcad, Proc. Third Intern. Symp. on NQR, V3Ilerini, Pisa (1975) p. I. [ 10 j P. Pyykko, Proc. Phys. Sot. (London) 92 ( 1967) 811. [IL] S. Rothcnberg, R.H. Young and H.F. Schaefer III, J. I+. Chcm. Sot. 92 f I9701 3243. [12] J.D. Petke and J.L. Whitten, J. Chcm. Phys. 56 (1972) 830.