Magnesium Photoionization: a K-Matrix Calculation with GTO Bases
R. MOCCIA
Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via Risorgimento 35, I-56126 Pisa, Italy
P. SPIZZO
Istituto di Chimica Quantistica ed Energetica Molecolare del CNR, Via
Risorgimento 35, I-56126 Pisa, Italy
1.Introduction
The theoretical study of the molecular photoionization processes is an active field of the current research. The calculation of the photoionization cross sections may be advantageously done by resorting to the use of bases. In fact, these bases are used by many powerful and available codes devised to treat bound-state problems, which may be adapted to consider also the electronic continuum. If only the integral cross section is required, the calculations are greatly simplified by methods, like the Stieltjes imaging (1), that allow to obtain these quantities without a detailed knowledge of the continuum wavefunctions. But if more detailed quantities are wanted, like the branching ratios, the differential cross sections, the structure due to narrow resonances etc., a more detailed knowledge of the continuum wavefunctions is necessary. In the atomic case, there exist efficient techniques for the continuum properties and may be applied in connection with the powerful CI packages that are currently available. Their application to molecular systems is thwarted by the re- quirement that the variational functions must be accurate in a region far away from the nucleus. Thus, they should be expanded upon bases of very diffuse orbitals, like the STO or STOCOS ones (2). These bases are very cumbersome for molecular cal- culations and only in few cases (usually the hydrides) one may tradeoff the problems in the multicenter bielectronic integrals for the well-known shortcomings of a monocentric expansion. To take a full advantage of the current packages for molecular structure, it is clearly necessary to develop an technique capable of extracting the continuum properties from the comparative short-range representations allowed by the GTO bases, which are an almost obligatory choice for molecules.
Unfortunately, in the molecular systems the theoretical predictions for the already formidable electronic problem cannot be checked fairly against the experimental data, since the nuclear motions may play major effects. From here the need to check these methods in calculations on atomic systems, where accurate theoretical and comparable experimental reference data are already available.
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plied to various atomic systems using STOCOS bases (3–5) and then to Helium with GTO bases (6). This last calculation has shown the capability of our K-matrix technique to obtain the continuum properties with GTO bases. As a matter of fact, accurate results were obtained also in the energy regions of the autoionizing states, where it is necessary to recover the interactions between diffuse discrete states and a continuum. The present paper applies this method to Magnesium and shows that it deals effectively also with other technical difficulties that are encountered in molecular calculations, e.g. the orthogonality to the inner shells and the strong short-range deviations from the Coulomb potential.
Unless otherwise specified, all quantities are expressed in atomic units.
2. Method of calculation
The present method is an extension of the K-matrix technique pioneered by Fano (7). Our previous works have discussed thoroughly its general aspects, the discretization procedure (4,8) and the implementation upon the short-range GTO bases (6), so only a concise description will be given here.
The K-matrix method is essentially a configuration interaction (CI) performed at a fixed energy lying in the continuum upon a basis of ”unperturbed functions” that (at the formal level) includes both discrete and continuous subsets. It turns the Schrödinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon a finite basis set.
In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour.
For conciseness, throughout this article it is understood that all the states and manifolds have well defined symmetry, so the corresponding labels
and projectors are omitted wherever this is possible without ambiguities.
2.1.THE K-MATRIX UNPERTURBED BASIS STATES
The formal basis employed in the K-matrix calculation includes the relevant partial wave channel (pwc) subspaces plus a ”localized channel” (lc) of discrete functions.
These last are usual CI states and their inclusion in the basis allows to efficiently reproduce the autoionizing states and the correlation effects.
In the atomic case, the pwc’s are defined by the ion level I and the l value of the electron partial wave, i.e. the formal pwc subsets span the tensor product of the ion level states times the one-electron l-wave manifold. In the following, the subspaces
will be indexed with greek letters; a subspace index
will designate explicitly
an open pwc subspace, while an index
an arbitrary subspace. The lc subspace will
be numbered 0 and
will denote the projector in the subspace
The formal basis employed for the pwc
is made by
the eigenfunctions
o
f the channel hamiltonian
i.e. the hamiltonian projected
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in the pwc subspace
The
are coupled products of the ion states times the an-
gular and spin functions of the outer electron (the so-called channel functions in the close-coupling jargon) and N is a normalization factor arising from the lack of strongorthogonality of the outer orbital to the ion states. The radial functions behave asymptotically as standing shifted Coulomb waves and the pwc basis functions may be indexed by the corresponding energy For ease of writing, the same notation is employed for both discrete and continuous eigenfunctions, normalized to unity and respectively.
Since these formal bases, which are supposed to describe the true continuum background, will be represented upon finite sets, all the quantities which must be interpolated from these representations (i.e. matrix elements and phaseshifts) must be smooth functions of the energy index: this requires a suitable redefinition of the
channel hamiltonian if this supports narrow shape resonances.
Using GTO bases, it cannot be expected that the variational representations of the electron waves are sufficiently accurate far outside the so-called “molecular region”, i.e. the rather limited region of space where the potential clearly deviates from the asymptotic Coulomb form. Therefore the phaseshifts of the pwc basis states cannot be obtained from the analysis of their long-range behaviour, as was done in previous works with the STOCOS bases. In the present approach, this analysis may be avoided since the K-matrix technique allows to determine, by equation [3] below, the phase-
shift difference between the
eigenfunctions
of
and the auxiliary basis functions
where
are the
bound and continuum radial eigenfunc-
tions in the Coulomb field. This requires variational representations accurate only where the potential felt by the outer electron is different from that of a pure Coulomb field and therefore this phaseshift difference builds up. Unless in special cases, the
auxiliary basis cannot be an orthonormal one.
2.2.THE K-MATRIX METHOD
If at the energy E there are n open channels, one may define n linearly independent trial functions of the form
where P denotes the principal value of the integral and the summation over the
discrete part and the integration over the continuous one of the subsets. The expan-
sion coefficients
are determined by imposing
and
this leads to the system of coupled integral equations
The delta-function addendum removes the divergences from these matrix elements and allows their representation upon bases. When the pwc basis functions are
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the eigenfunctions of the projected hamiltonian
the intrachannel matrix
elements
(E) are identically zero, but this is not required to apply the method
and indeed it must not hold for the auxiliary basis
The K-matrix on
the energy-shell K ( E ) , defined by
is a real
symmetric matrix on the real energy axis. As discussed in (4), it is related to the scattering matrix and contains the quantities needed to analyze the resonances.
It should be noted that the integral equations [2] determining the elements
derived as an energy-variational problem, correspond also to the stationary condition of the variational functional proposed by Newton (9). Thus the K-matrix elements obeying equation [2] guarantee a stationary value for the K-matrix on the energy
shell.
When only one channel is open, the phaseshift
is related to that
of the unperturbed basis function
) by
This relation allows, as said above, to obtain the phaseshifts
of
the
basis
func-
tions
by a single-channel K-matrix calculation on the basis
, whose non-
Coulomb phaseshifts are zero by construction.
The real K-matrix variational wavefunctions
satisfy
so that two sets of complex orthonormal eigenfunctions may be obtained by:
where
is
the total
phaseshift of
.
The
states
obey
the boundary conditions suitable for
photoionization
processes,
since
they contain an outgoing Coulomb wave (with zero phaseshift) only in the channel
As discussed in (4), the K-matrix has a pole at energies near a resonance and this yields a convenient method for the analysis of the narrow autoionizing states. The matrix representation of equation [2] upon a finite basis may be in fact recast in the form (4)
where P ( E ) is an almost diagonal matrix arising from the integration of P/(E – E') times the polynomials that interpolate the K- and V-matrix elements over the basis grid. Across a narrow resonance, the real symmetric matrix should be a smooth function of the energy and one of its eigenvalues must change sign. Its inverse may be therefore approximated, quickly and accurately, using the smallest (in modulus) eigenvalues and their eigenvectors, which may be linearly interpolated across the resonance. The blocked and almost-diagonal matrix P ( E ) may be easily inverted, so it is possible to sweep the resonance profile with a great saving of CPU time.
This approach proved accurate and convenient for the analysis of the narrow resonances; the results presented in this work have been obtained without employing this trick because the limited dimensions of this single-channel problem are easily handled
by the standard method.
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2.3.THE BASIS FUNCTIONS
The present calculations are at the frozen-core level; the inclusion of a phenomenological core potential is straightforward and has been avoided here only because it will complicate the comparison of the results on GTO and STOCOS bases. All the basis functions have the form is a single Slater determinant built with the SCF orbitals for the ground state; for simplicity the core will be hereafter omitted. The strong-orthogonality to a closed shell SCF core does not cause any loss of generality and has been imposed for computational ease on the valence group functions i. No other orthogonality constraint is imposed on the basis functions, in particular the waves are not strong-orthogonal to the ion states nor the pwc basis functions are mutually orthogonal. Each of these conditions, beside slowing the convergence of the variational expansion, would inhibit the phaseshift determination by the K-matrix calculation on the auxiliary basis.
The core orbitals have been expanded upon an 11s/5p GTO basis and the group energy of the core is –198.747 against the value –198.823 obtained with a 4s/3p
STO basis. As well known, very long gaussian expansions are needed to obtain inner orbitals of near-HF (SCF limit) quality, which is instead reached by relatively short
STO expansions. The GTO core employed here yields an higher total energy, but generates a more attractive potential and hence more negative attachment energies
for the outer electrons.
The 3s and 3p orbitals of
have been expanded upon all the GTO employed
for the inner orbitals plus other 5 GTO whose orbital exponents were optimized for them.
For the description of the 3sel continua and of the higher 3pnl resonances, the bases
include a large number of configurations of the forms
, where 3.s, 3p
are the lowest one-electron states in the field of the SCF
core.
The localized basis function for the set 0 (ls) are usual frozen-core valence-shell CI states; all the bound states involved in the present calculations are also described at this level.
2.4.THE REPRESENTATION OF THE PARTIAL WAVE CHANNELS
The
epresentations
of the unperturbed states
(uppercase
letters are used for the formal basis and lowercase ones for their
representations)
have been obtained by diagonalizing the electrostatic hamiltonian over the
basis configurations mentioned above.
The variational representations
of the regular Coulomb waves have been obtained
by diagonalizing the Coulomb hamiltonian upon the same orbital basis employed to expand the waves . The following discussion refers explicitly to the states
but apply equally well to this ones.
The basis orbitals employed for the electron wave have the form
where is a third-degree polynomial in with coefficients depending on The present work is mainly aimed at demonstrating the ultimate accuracy of the method, so the sequence of the orbital exponents and the coefficients of
were determined to yield a large number of variational states in the lowest continuum
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R. MOCCIA AND P. SPIZZO
with the least redundance of the metric. In atomic calculations, the rather expensive introduction of these polynomial factors appears to be fully justified only in rather special cases. Indeed, here one deals generally with a rather limited number of open channels and may therefore employ a large number of basis orbitals, so a low metric redundance is the only practical advantage. As a matter of fact, results of comparable quality were obtained in preliminary calculations without this factor. The molecular calculations, instead, require the proper consideration of many partial wave channels, so the choice of the above parameters may be used to minimize the number of basis functions.
The variational pwc states obtained with these bases include accurate representations for the lowest bound states of the channel hamiltonian, broad ”wavepackets” in the higher Rydberg region, a number of narrow wavepackets in the lowest continuum and again broad wavepackets at higher energy. In this context, narrow wavepacket means a variational state whose wave is almost exact, i.e. , inside a sufficiently large sphere. When, as in the present case, the channel hamilto-
nians do not support shape resonances, the energy-normalization constants of these narrow wavepackets may be fairly well approximated from the energy spacings,
The present method does not involve the analysis of the long-range behaviour of the states, so its application requires only that the narrow wavepackets are accurate inside the molecular region. By equation [3], the phaseshifts of these states may be determined through a K-matrix calculation on the auxiliary basis, so it is assumed that the narrow wavepackets might be continued outside the molecular region as shifted Coulomb waves.
The K-matrix calculations may be obviously performed only at energies inside the range covered by the narrow wavepackets, which should allow to interpolate the matrix elements and the phaseshifts of the channel basis functions. The contributions to the integrals in equations [1,2] from the region of the narrow wavepackets are obtained by interpolating the integrands on the grid supplied by the variational basis. Those from the high-energy regions, which should be small and weakly energy dependent, are instead approximated by summing the contributions of the broad wavepackets.
The
basis for the discretized K-matrix calculations contains the narrow wavepack-
ets in
their ”energy-normalized” form
and all the other variational states
with unit normalization.
The calculations performed with gaussian bases have been checked employing also the STOCOS bases which include, beside Slater and Hydrogenic orbitals, a large number of STOCOS functions
These are probably the most efficient bases for calculations in the electronic continua of atomic systems and furnish reliable reference data for the GTO calculations.
3.Results
The present work is essentially concerned with the comparison of the results obtained with the GTO and STOCOS bases: a thorough comparison of the STOCOS results with the experimental data has been given in (3).
As noted above, the field exerted by the present GTO core on the outer electrons
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R. MOCCIA AND P. SPIZZO
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is slightly more attractive than that of a near-HF core and the attachment energies for the outer electrons are slightly more negative than those obtained in valenceanalogous calculations with STO bases. This partially compensates the error due to the intershell correlation, so, by a fortunate cancellation of effects, the GTO calculations yield valence energies in better agreement with experiment than those carried out with STO bases; the most significant positions are reported in table 1. Also the transition energies, of course, turn out slightly better with the GTO basis than with the STOCOS one: the transition is predicted at 0.15770 and 0.15715 with the GTO and STOCOS bases against the experimental value 0.15970.
In spite of the relatively poor description of the core, the present GTO calculations yield accurate oscillator strengths: the data for the lowest-energy transitions from the Mg ground state are given in table 2 and compare well with the STOCOS results.
The experimental values given in this table represent a less relevant comparison, due to their uncertainties and/or normalizations upon theoretical results (3). All the transition probabilities have been computed with both the length gauge (LG) and the velocity gauge (VG) forms of the dipole operator and the gauge invariance of the results is only slightly worse than that achieved with the STOCOS bases.
The properties of the
continuum and the ground state photoionization cross sec-
tion have been studied from the 3s threshold up to a wave energy of about 0.150. From this energy to the 3p ionization threshold at about 0.157, the crowding of the resonances of the 3pns and 3pnd series makes hopeless further variational calculations. The quality and regularity of the present results, however, allow to extrapolate safely the properties of this region, e.g. by fitting formulae based on the quantum
defect theory.
In addition, we have investigated the broad
resonance, which lies rather close
to the ionization threshold and represents therefore a stringent test for the capabilities of the method in the delicate low-energy region.
The positions and the widths for the above autoionizing states have been obtained from the analysis of the scattering matrix as described in (4); the results of the present
GTO and STOCOS calculations are reported in table 3. On the whole, the GTO and
STOCOS results compare quite well, with a significant difference only for the width of the 3p3d resonance. As discussed in (3), a frozen-core calculation of this kind reproduces accurately the position of the levels 3snl, 3pnl with respect to their parent ion. Indeed, the calculated wave energies of the 3pns and 3pnd resonances compare well with their experimental counterparts when corrected for the intershell energy difference 0.0054 between the 3s and 3p ion levels. Instead, the position of the resonance cannot be reproduced very accurately with respect to neither of these thresholds, since its intershell correlation energy is intermediate between theirs. The GTO and STOCOS calculations yield values in good agreement for this resonance, although the predicted width is somewhat larger than the experimental value
The ground state photoionization as calculated with the GTO and STOCOS bases is given in figure 1. The agreement of the two calculations is satisfactory, apart in the critical region of the very low wave energies. It is interesting to note that the correlation effects in this continuum are so strong that the cross section to the unperturbed pwc basis functions (which are the best frozen-core single-configuration approxima- tions) is far from being the coarse average of that to the correlated states: it amounts to 10 Mb at the threshold and would lie off the figure in most of the energy range.
The phaseshifts for the manifold are reported in figure 2 from the ionization threshold up to the photoelectron energy 0.1, across the lowest broad resonance
