Experimental investigation of small He clusters

(1)

Experimental investigation of small He clusters

Citation for published version (APA):

Deursen, van, A. P. J., & Reuss, J. (1975). Experimental investigation of small He clusters. Journal of Chemical Physics, 63(10), 4559-4560. https://doi.org/10.1063/1.431139

DOI:

10.1063/1.431139

Document status and date: Published: 01/01/1975

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

(2)

Letters to the Editor 4559

13Data deconvolution alters the excitation function only quanti-tatively, not qualitatively. The complete data and their analy-sis in terms of a two channel model will be published else-where.

14J. N. Bardsley, J. Chern. Phys. 51, 3384 (1969).

158. J. Nalley, R. N. Compton, H. C. Schweinler, and V. E. Anderson, J. Chern. Phys. 59, 4125 (1973).

16R. D. Levine, R. B. Bernstein, Molecular Reaction Dynam-ics (Oxford University, Oxford, 1974).

17p. J. Chantry, J. Chern. Phys. 51, 3380 (1969).

Experimental investigation of small He clusters

A. P. J. van Deursen and J. Reuss

Katholieke Universiteit, Fysisch Laboratorium, Toernooiveld, The Netherlands

(Received 16 December 1974; revised paper received 2 September 1975)

The question of the existence of a bound van der Waals complex for Hea has received attention in the literature recently. Accurate scattering studies by Farrar and Lee! and by Bennewitz et al. a suggest the absence of a bound state for the Hea system. The potentials obtained are in fair agreement with theoretical results by Bertoncini and Wahl3 and by Schaefer et al. 4 How-ever, as is discussed by Bennewitz, 2 a 1. 4% increase

in the potential depth suffices to accomodate one bound state with zero angular momentum.

In the course of measurements of the production of molecular clusters5 _{we have searched for helium}

clus-ters. In our molecular beam machine we are able to cool the source to 6 K. The source pressure Po can be varied up to 6 atm for a nozzle diameter of 6 IJ.. The beam is detected by a magnetic spectrometer.5 _The

ionizer is adjusted to work without any detectable space charge.

The measured intensities of the ions He:, n

=

1-4 are given in Fig. 1 (a) as a function of Po, at To = 7 K. He; ions are observed for values of Po larger than 200 torr. The intensity of He+ increases linearly with Po for Po < 400 torr, and the intensity of He~ varies with

1'3

up to

Po

= 800 torr. Above 1100 torr the He; and He~ ions appear; their intensity is roughly pro-portional to P~o. The heaviest ion detected at 1500 torr was Hei,.

To determine the correspondence between cluster ions and the neutral parent clusters, a scattering cham-ber filled with argon gas at 80 K is placed in the neutral beam path. A similar experiment with argon clusters is described in Ref. 6. The effective total cross sec-tion for the different ion signals is given in Fig. lb.

The cross section of the He+ signal is constant over the entire pressure range. We conclude that the He+ ions are due to the unclustered He atoms in the neutral beam.

The cross section of the He· signal is constant for inlet pressures up to 500 torr. In this pressure range one neutral parent cluster for the He; ion exiSts.

From three arguments we conclude that the He; ions originate from neutral Hea clusters for Po between

200 and 500 torr, which dissociate to Hez in the ionizer.

Firstly, the intensity is proportional to

1'3.

For the Hea parents a proportionality of ~ is expected.

Secondly, the ratio of the total cross sections for He+ and He; is found to be as large as 1. 95 ± O. 1. It

is difficult to reconcile this large ratio with a Hea parent. 6

Finally, the intensity of the trimers relative to the

intensity -TI--r--,-...,-..-.-, ,..., ,..., ~I --~ cross-section -rTTTT-r----,

in arb. units in arb. units

103 9 6 5 4 3 Z 102 9 6 5 4 3 101 9 6 5 4 3 2 I 11[ _Ii 2 3 4 5 679 103

z

.50 .40

II

.30

/

.20

_...

_/"

i

][ I-I-'-'-f-'" .10

**_{. __ ._--*----_.}**

3 4 5 678 103 2 Torr

FIG. 1. Intensity and cross section vs inlet pressure. Curves I, II, ITI, and IV correspond to ion masses 4, 8, 12, and 16, respectively.

(3)

4560 Letters to the Editor

monomers can be estimated6 _{assuming equilibrium in}

the source and assuming one bound state for a trimer. Good agreement is found with experimental values for a trimer parent; for a dimer parent, however, the estimated concentration is too large by an order of magnitude.

At pressures larger than 500 torr the cross section on He~ rises owing to admixture of larger clusters in the beam and dissociative ionization. The He'S cross section at Po > 1000 torr varies strongly with Po and is about 3.5 times larger than the He cross section.

It seems impossible to link this behavior with a unique parent cluster.

NOTES

In conclusion, our results agree with the prediction that there is no bound state for Hea•

lJ . H. Farrar and Y. T. Lee, J. Chern. Phys. 56, 5801 (1972).

2H• G. Bennewitz, H. Busse, H. D. Dohman, D. E. Oates, and

W. Schrader, Z. Physik 253, 435 (1972).

3p • Bertoncini and A. C. Wahl, Phys. Rev. Lett. 25, 991 (1970) and J. Chern. Phys. 58, 1259 (1973).

4H• F. Schaefer, D. R. McLaughlin, F. E. Harris, and B. J. Adler, Phys. Rev. Lett. 25, 988 (1970). D. R. McLaughlin and H. F. Schaefer, Chern. Phys. Lett. 12, 244 (1971). 5 A. van Deursen and J. Reuss, Int. J. Mass Spectrorn. Ion

Phys. 11, 483 (1973).

6A. van Deursen and J. Reuss, Int. J. Mass Spectrorn. Ion Phys. (in press).

Pseudopotential transformation of correlated-pair equations

Levente Szasz and Lawrence Brown

Department of Physics, Fordham University, New York, New York 10458

(Received 16 June 1975)

In the theory of correlated wavefunctionsl_-3 _the

cor-related wavefunction of a pair of valence electrons is de-termined from the equation:

(1)

where Hi and Ha are the Hartree-Fock Hamiltonian op-erators for the N core electrons expressed in terms of the coordinates of the first and second electron, V 12 is

the Coulomb interaction, and Pia is the projection op-erator orthogonalizing any arbitrary two-particle func-tion to the core orbitals CP1 ••• cP N'

Let us define a general pseudopotentia14 _{as follows:}

N

Vd(l)=~CPs(l)

f

F s(1')f(l')d1', (2) where Fs is an arbitrary function, and let us consider the equation

(3 )

We state that the energy spectrum of Eq. (3) is iden-tical with the spectrum of Eq. (1). We recall2

•3 that Pia

has the form P₁₂

=

P_{1P 2}where P_{1 ;;}(1 -

n

₁₎and

N

nd(l) =

L:

CPs(l)

f

cp':(l')f(l')d1'.

s=l

(4)

It is easy to see that P₁₂Vi

=

0, (i

=

1, 2, ) and P 1aHj

=

HjP_12,

(i = 1, 2). Operating from the left on both sides of Eq.

(3) by P₁₂we get

(5 ) By writing <I>~ =P12-¥~we obtain from Eq. (5) that <1>: sat-isfies the equation

(6)

However, the operator in the bracket of Eq. (6) is the same as in Eq. (1); therefore we can state that <I>~

=

<l>k

and €k =Ek' Q. E. D. (We note that writing <1>: in the

form of P 12-¥k does not restrict the generality of the

so-lution since the soso-lutions of Eq. (1) must be orthogonal to the core orbitals. It is the operator P₁₂which orthog-onalizes an arbitrary function to the core orbitals. )

The main point in this transformation is that the solu-tions of Eq. (3) are pseudowavefunctions, i. e., they do not have to be orthogonal to the core functions. While the eigenvalues of Eq. (3) are identical with the eigen-values of Eq. (1) the approximate solutions of Eq. (3)

will be much simpler to compute than the solutions of

Eq. (1). The transformation constitutes a

simplifica-tion without the loss of accuracy.

Another advantage of the transformation is the arbi-trariness of the pseudopotentials. This enables the in-troduction of a Simple model potential3.5 in a less arbi-trary fashion than was done before. One can introduce the model potential Vii as an approximation to the Har-tree-Fock potential plus the pseudopotential and treat the deviation as a perturbation. This procedure leads to the equation

II iI ) - A A (7)

(Hi +Ha+V12P12 -¥k=€k-¥k'

where H~ '" - ~ ~ + Viii. If V AI is a good choice, Eq. (7)

will have a very simple mathematical structure and

Ek

will be a very good approximation to the exact eigenval-ue €k'

Finally we note that the transformation presented above can be applied to any correlated-pair equation e. g., to the general pair equation of two core elec-trons.3 _{Using the transformation a new, simplified}