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Approximate Molecular Orbital Theory for Positrons
Approximate Molecular Orbital Theory for Positrons and Positronium Atoms Bound to Moleculest D. M. Schrader* and C. M. Wang Chemistry Department, Marquette University, Milwaukee, Wisconsin 53233 (Received November 12, 1975) Publication costs assisted by the Petroleum Research Fund
An approximate molecular orbital theory for bound positrons is presented. The theory is semiempirical and the parameterization scheme is based upon some very sparse data together with plausible interpolation ideas. We follow the CNDO/2 approach for the positron, and use that method also for the purely electronic part of the theory; molecules comprising first-row atoms are considered. The calculations yield fully selfconsistent positronic and electronic molecular orbitals, as well as positron and positronium affinities of molecules. Results for over 60 molecules presented, among which are eight studied by Madia et al. who used a combination of ideas to construct a semiempirical approach for the positronic molecular orbitals; we compare our results with theirs for the eight common molecules. General features of positron and positronium binding are discussed. Positronium appears to be stable to split-off in molecules in which it replaces hydrogen; for weak acids, a good correlation is found between acid strength and positron affinity for the anion; however, the positron does not appear to be a typical electrophile in aromatic addition reactions. Some justification of our approach is provided by agreement or consistency of our calculated results with some recent experimental observations: our calculated positronium affinities for toluene and nitrobenzene (0 and 1.1f 0.8 eV, respectively) are in excellent relative and fair quantitative agreement with enthalpies of positronium attachment for these molecules observed by Ache and co-workers (0 and 0.20 f 0.01 eV, respectively) and with the positronium binding energy of nitrobenzene obtained by Goldanskii and co-workers by fitting experimental data to a simple model potential (0.15 to 0.17 eV, depending upon solvent); our calculated positronium affinity for p -benzoquinone is in disagreement with observations by these two authors, however. For smaller molecules, positronium bond strengths in PsOz, PsNO, and PsN02 (0, 0, 1.95 eV, respectively) seem to be consistent with relative tendencies of gaseous 02, NO, and NO2 molecules to quench triplet positronium by chemical combination as opposed to spin conversion (0, 0, 25%) as reported by Goldanskii and co-workers. Quenching cross sections for these three molecules as reported by Chuang and Tao (0.002 65, 0.008 45, and 26 A2, respectively) seem to suggest the stability of PsNO2 but not Ps02 and PsNO, in agreement with our calculations.
I. Introduction Positron annihilation in a molecular medium proceeds by a complicated mechanism featuring a number of competing pr0cesses.l While all these ultimately lead to the same result for the positron, namely, annihilation, the characteristics of the final state of the host molecule, as well as the positron lifetime and angular correlation of annihilation radiation, depend strongly upon the history of the system. In modern experiments, several competing positron decay channels can be distinguished and partly characterized. In order to explain observed positron lifetimes and angular correlation of annihilation radiation, experimentalists have presumed a variety of decay channels: annihilation from bound states of positron-molecule complexes, bound states of positroniummolecule complexes, scattering states, as well as resonance states. The importance of bound states of positronium-molecule complexes has been reviewed recently by Goldanskii and Shantarovich.2 + Presented in part a t the VI11 Midwest Theoretical Chemistry Conference, Madison, Wisc., May 1-3, 1975; and a t the IV International Conference on Positron Annihilation, Helsingdrs, Denmark, Aug. 23-26,1976.
* Address for 1976-1977 academic year: Mathematics Department, University of Nottingham, Nottingham NG7 2RD, United Kingdom.
It seems worthwhile to supplement these attempts a t rationalizing observed results with quantum mechanical calculations on the intermediate complexes. The purpose of this paper is to present a molecular orbital theory for the structure of bound states of positron- and positronium-molecule complexes. In this first attempt, the important short-range electron-positron correlation effects3 are merely parameterized. The highly empirical CNDO theory4 is used for the electrons, and the positronic molecular orbital is calculated using a set of assumptions which are similar to the CNDO approach as available data permits. The CNDO assumptions for electronic systems correspond to approximations which appear to be gross. Yet the method has been highly successful in applications on purely electronic molecules, not so much in absolute predictions as in relative predictions and rationalizations of chemical data for sets of related molecules. The reasons for this success remain something of a mystery. We hope that, by using the same approach for the positron, our calculations will also be useful for relative predictions and rationalizations of results for related molecules. However, we should not expect accurate numerical determination of positron interaction parameters with individual molecules. A great advantage of a CNDO-type approach is its simplicity: large molecules can be routinely treated. We hope, by looking a t a large number of related molecules, to discover
The Journal of Physical Chemistry, Voi. 80, No. 22, 1976
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D. M. Schrader and C. M. Wang
trends in positron binding mechanisms which will give rise to some fundamental understanding of the structure of such systems. We believe that the present work makes some progress in that direction. It should be noted that recently5 a theory was proposed for positronic molecular orbital computations. That theory is like our own in that the CNDO/B approach was used for the electrons; however, a combination of approximate ideas was used for the positron. No empirical information was used to evaluate positronic parameters in that theory. While experimental information on positron binding is very sparse, it is not nonexistent; all such information pertaining to atoms and diatomic molecules known to us was used to calibrate the approach described in the present work.
11. CNDO Theory for Positrons Ignoring for the moment electron-positron correlation, we can write an approximate molecular orbital wave function for the system consisting of a positron bound to a closed shell electronic system as6
Ffii = H,,
- cc PXu(fii1Aa) X u
(6)
where H and H, are the matrices of the core Hamiltonian:
where VA is the potential generated by the nucleus and core electrons of atom A. (We neglect a very small difference in VA for electrons compared to positrons due to exchange.) The matrices P and P, in eq 6 are the density matrices
P,, = 2
occ
C c;i
cui
1
P,i = c;, cup and the symbol (pv1Xu) denotes the interaction integral $, is the molecular orbital for the positron (positronic molecular orbital, PMO), $i are the electronic molecular orbitals (EMO), and .A is the n-electron antisymmetrizer and normalizer. Each molecular orbital is written as a sum of atomic orbitals:
Each electronic basis function 4, is a Slater-type atomic orbital characterized by quantum numbers n, 1, and m , and an exponential parameter {, The functional form for the STO's is familiar to many; it is given, along with values for {, on pp 23-29 of Pople and Beveridge's book.4 The basis for the PMO, I$;), we discuss below. In general, a bar above a subscript or the subscript p in this work denotes a positronic quantity. Minimizing the expectation value of the Hamiltonian by varying the coefficients in eq 2 while keeping the MO's normalized and the EMO's orthogonal leads to Roothaan-Hartree-Fock7 type equations:
(F - qS)ci = 0
i = 1,2,...,
(F, - E,S,)C~ = 0
(3)
which are coupled together and are solved self-consistently. c; and c, are column vectors of coefficients for the i t h EM0 and the PMO:
The symbol (jZlXu) denotes an electron-positron interaction integral; its definition follows from eq 10. In eq 6, the signs attached to these quantities should be noted, along with the absence of an exchange term in the second part of eq 6. The zero differential overlap approximation is applied to the positron in the same way as to the electrons: (11)
This approximation is applied consistently throughout the CNDO approach except in the evaluation of those matrix elements of H and H, (eq 7 )which contain a two-center charge distribution in the integrand. These integrals are responsible for most of the chemical bonding effects. Rather than neglecting them, they are set equal to an empirical parameter:
I
H,u = P,,] P # v, Ir on A ] (12) v on 0, A # B H,, = P,, Rotational invariance is automatically achieved if all the integrals that appear in the Roothaan-Hartree-Fock equations (eq 3) are evaluated by integration. If some integrals are approximated, as in CNDO theory, rotational invariance must be endowed by subsidiary conditions. Our choice of conditions for the positron is exactly the same as Pople's for the electrons:
I -
(4,l V B I ~ , ) = VAB,(+E( V B I ~ / I=) VAB,fi on A (PWIV V ) = TAB, (LEI vv) = TAB, v on B 4,, A,, 4ii and t, are the eigenvalues for the ith E M 0 and the PMO, F and S are the electronic Fock and overlap matrices, and F, and S, are the corresponding positronic quantities: ti
The Journal offhysical Chemistry, Vol. 80, No. 22, 1976
The parameters p,," and 0~; are set equal to
?C B
s type
(13)
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Approximate Molecular Orbital Theory for Positrons where S,, and Sj;are calculated, and P O A and Pox are bonding parameters characteristic of atom A, not the individual orbitals on A. These approximations render the Roothaan-Hartree-Fock equations (eq 3) as follows:
(F - e L l ) c = L0
i = 1,2,.
..
(FP- tPl)cp= 0
is quite different from that of Pople: we must fit our five numbers to some orderly and sensible scheme which permits us to interpolate values for lithium through nitrogen. To aid us we must make some assumptions regarding the positron affinities of the first row atoms. These assumptions can be modified later as more experimental and theoretical data become available. At this time, we do not have enough data to enable us to average our positronic parameters over multiplets as Pople did for the electronic calculations. We must be content with what little ground state data on positron-atom binding is presently available. Let us consider the diagonal matrix elements first. We may write, from eq 15, 16, and 18:
F~~ =
1
F, = 2S ~ ( P O + A Pod
p #
v
(15)
where I.L is on A and v is on B. U,, and U;p above are the atomic integrals
wpfi - PAAYKA)
+ (Bc #A
Q B ~ ~ B )
1
(hl- 2 v 2 + VAI$b)
(16)
and PBB (P& is the electronic (positronic) population of atom B:
PBB= C P,, u
on B
PBB=
uonB
P,;
(17)
We use Pople's values for the electronic integrals in eq 15, and adhere as closely as possible to his methods for arriving a t these in assigning values for the corresponding positronic integrals. For example,. VAB is taken to be TAB times the number of valence electrons on atom B, ZB:
VAB= ZBYAB
VAB = ZBYAB A # B
(18)
and Sobin eq 15 are evaluated by direct integration, as are their electronic counterparts. Pople chooses values for P O A which cause the CNDO results to mimic those for minimal basis ab initio calculations. No such calculations exist to guide us in choosing values for Pox, so we resort to a semiempirical determination described below. The atomic integral U,, can be related to the ionization potential I , and to the electron affinity A,. Pople ignored exchange contributions to write an average expression as
I, and A , are taken directly from atomic spectral data as averages of atomic term values weighted according to the multiplicity of each term arising from the lowest configuration. A great deal of such data exists for parameterizing the electronic quantities, but positron ionization potentials and affinities are generally unknown. Indeed, the literature shows only five usable numbers of our parameterization! These are: the positronium affinities of H (1.014 eV, ref 8), 0 (2.2 f 0.5 eV), OH (
