Cent. Eur. J. Phys. • 8(5) • 2010 • 825-832
DOI: 10.2478/s11534-009-0162-1
Central European Journal of Physics
Non-physical consequences of the muffin-tin-type
intra-molecular potential
Research Article
Miron Ya. Amusia12∗ , Arkadiy S. Baltenkov1
1 Racah Institute of Physics, the Hebrew University, Jerusalem 91904, Israel
2 Ioffe Physical-Technical Institute, St. Petersburg 194021, Russia
3 Arifov Institute of Electronics, Tashkent, 100125, Uzbekistan
Received 7 August 2009; accepted 7 November 2009
Abstract:
We demonstrate, using a simple model, that, in the frame of muffin-tin-like potential, non-physical peculiarities appear in molecular photoionization cross-sections that are a consequence of “jumps” in the potential
and its first derivative at some radius. The magnitude of non-physical effects is of the same order as the
physical oscillations in the cross-section of a diatomicmolecule. The role of the size of these “jumps” is
illustrated by choosing three values for it.
The results obtained are connected to the previously studied effect of non-analytic behavior as a function
of r, the potential V (r) acting upon a particle on its photoionization cross-section. In reality, such potential
has to be analytic in magnitude and have a first derivative function in r. The introduction of non-analytic
features in model V (r) leads to non-physical features – oscillations, additional maxima, and so forth – in
the corresponding cross-section.
PACS (2008): 31.15.xr, 33.15.-e, 33.80.Eh
Keywords:
muffin-tin potential • barrier penetration • molecular photoionization
© Versita Sp. z o.o.
1.
Introduction
Almost all nuclei, atoms, molecules, clusters, and macroscopic bodies consist of a number of particles. Since the
respective Dirac and Schrödinger equations that account
for the inter-particle interactions describing these objects
cannot be solved precisely enough, even by means of the
most advanced computers, approximations are unavoidable
and simplifications are inevitable.
∗
E-mail: [email protected]
One of the most popular simplifying approaches is the socalled mean field approximation. In its frame, it is assumed
that inter-particle interaction can be taken into account
sufficiently accurately by a choice of mean field acting
upon constituent particlesnucleons in nuclei, electrons
in atoms and molecules, and so forth.
Best known is the Hartree-Fock (HF) approximation. It
is good and reasonably easy to solve for atoms and nuclei. But, even for many-particle atomic objects that are
not too complex, a simplified version of HFthe HermannSkillman potential [1] is used. For molecules, muffin-tin
type potential [2] is commonly employed. In solid bodies,
along with muffin-tin potentials (MTP) [3], corrections are
825
Unauthenticated
Download Date | 6/18/17 10:38 AM
Non-physical consequences of the muffin-tin-type intra-molecular potential
often introduced in order to improve the mean potentials
that eliminate the self-action of electrons (see, e.g. [4]). In
all the mentioned approaches, the one-particle potential
V (r) (as a function of the distance from the center of the
system under consideration) is non-analytic: the magnitudes and/or first derivatives have discontinuities.
As was demonstrated long ago in connection to the
Hermann-Skillman potential [5]1 , the discontinuities manifest themselves in spurious, entirely non-physical oscillations in, for example, the photoionization cross-section.
This effect was reanalyzed later in a number of papers [6–
8], leading to essentially the same results. Quite recently, it was demonstrated that the crude elimination of
self-action dramatically affects the photoionization crosssection of clusters [9].
The problem of a potential’s non-analyticity is far from
trivial. Contrary to widely –held belief, a singularity in
the potential does not lead one-to-one to a singularity in
the wave function and, correspondingly, in the singularity in the corresponding cross-section. To illustrate this
point, recall that, for a hydrogen atom with its singular as
1/r Coulomb potential, the photoionization cross-section
is actually a smoothly decreasing power-like function from
the very threshold.
In this paper, using a simple model, we will consider the
role of the muffin-tin potential’s discontinuity in the photoionization cross-section of diatomic molecules. As far as
we are aware, the consequences of this discontinuity have
not yet been addressed in the literature.
2.
Muffin-tin potential
As mentioned above in the Introduction, muffin-tin potential (MTP) is the theoretical construction that is widely
used in solid-state and molecular physics. Within the
framework of this approach, the potential of multi-atomic
systems is represented as a cluster of non-overlapping
spherical potentials centered on the atomic sites. In the
space between the atomic spheres, the potential is assumed to be constant. In solid-state physics, MTP covers
all space. Therefore, there is no question about the potential behavior beyond the microscopic body [10]. For
the molecular case, the situation is quite different. MTP
here is created by a finite number of atomic spheres, and
It was also demonstrated in [5] that the finite range of
numerical integration over radii that substitutes infinite
limits serves also as a discontinuity that is indeed reflected by non-physical oscillations in the photoionization
cross-section at high photon energies.
1
it is impossible to neglect the existence of the molecular
boundary. The adaptation of MTP for this case consists of
introducing a molecular sphere [2] that surrounds all atoms
forming the molecule, thus creating a sort of resonator for
the outgoing photoelectron waves.
Outside the MTP sphere, the photoelectrons experience
the pure Coulomb potential of the vacancy created in the
photoionization process. The center of this Coulomb potential is the center of the molecular sphere. This adaptation of MTP (the MTP-model) was suggested in [2] and is
currently widely used in calculations of molecular continuum wave functions and molecular photoionization (see,
e.g., [11–15] and references therein).
It is easy to see that, in the MTP-model, the potential of
the charged molecule on the surface of a molecular sphere
“jumps,” forming a potential barrier. Partial reflection of
the photoelectron wave emitted from the center of an atom
by this potential barrier inevitably leads to an alteration of
the wave function inside the atomic sphere where the matrix element of the photo-effect is formed. In sphericallysymmetric cases [16–18], this phenomenon leads to oscillations in the frequency dependencies of the photoionization
cross-sections of atoms located inside the potential cavity
sphere. The amplitude and period of these oscillations
strongly depend on the radius of the “resonator” and on
the coefficient of reflection of an electronic wave near its
walls.
The physical origin of these resonances is the same
as in the extended x-ray absorption fine structure (EXAFS) [19, 20]. Kronig’s short-range order theory [21, 22]
explains EXAFS by the modulations of the photoelectron
wave function in the final state. The difference between
the two cases is that the source of the reflected waves
in the EXAFS phenomenon is the nearby atoms of the
crystal, while, in the MTP-model, the source of backscattering waves is a potential barrier that also modifies the
frequency dependencies of molecular photoionization.
Hence, in the MTP-model [2, 11], natural diffraction effects
caused by the multi-center character of the molecular photoionization problem somehow combine with artificial barrier effects. The MTP sphere is an imagined design, and
its radius Rm is, generally speaking, chosen arbitrarily:
any sphere with radius R > Rm also can be considered
an MTP border. Thus, it is clear that the results obtained
in the MTP frame always include some uncertainty connected to the choice of the molecular sphere radius. This
radius also determines the height of a potential barrier on
the border of the MTP sphere. As far as we know, these
features of the MTP-model never attract attention, and
artificial barrier effects have not been investigated at all.
It is the purpose of the present article to clarify to some
extend the role of these features in the formation of con-
826
Unauthenticated
Download Date | 6/18/17 10:38 AM
Miron Ya. Amusia, Arkadiy S. Baltenkov
following qualitative picture. Inside atomic sphere I1 , the
potential is close to the potential of a positive ion. On the
border of the atomic sphere, the curve V (r) exhibits a break
at the transition to a constant. Between points A and B on
line 1, the molecular potential is constant. Then (at point
B) the MTP potential “jumps” at the molecular sphere,
changing from V (Ra ) to −1/Rm 2,3 . Examining other lines
of a trial charge motion, we conclude that the MTP-model
potential looks like a step of constant height, but varying
width. So, for trajectory 2 nearest the atomic center point
of connection of the atomic and molecular spheres, the
width of the potential step is zero.
Figure 1.
Muffin-tin potential for a molecule of two identical atoms.
tinuum molecular wave functions. Using the example of
spherical models, we analyze both the consequences in
calculating continuum wave functions as well as the photoionization cross-sections of “jumps” in potential and its
derivative.
The plan of the article is as follows. In Sec. 3 we analyze types of peculiarities in the MTP potential. Then,
in Sec. 4, we consider a spherically-symmetric atom-like
system with a potential containing a similar feature, and
derive the equations for calculating continuum wave functions in this model problem. The obtained general formulas will be used in Secs. 5 and 6 for numerical calculations
of the wave functions. The parameter that is typical for
the MTP-model will be used in these calculations.
3.
Details of the MTP model
According to paper [2], “The molecular field is defined by
a potential consisting of three types of regions, defined
by a set of non-overlapping spheres.” There are spherical
regions I1 and I2 where atomic potentials are assumed to
be spherically symmetric (Fig. 1). Outside these spheres
but inside the molecular one (region II), the molecular potential is assumed to be constant. Outside the molecular
sphere for ionized molecules, the molecular potential is
the Coulomb potential that originates from the center of
the molecular sphere.
Consider the behavior of the potential in which moves the
photoelectron that is eliminated after photon absorption
from a deep level of, for example, first atom I1 . If to move
a trial charge along line 1, as shown in Fig. 1, counting
coordinate r from a nucleus of this atom, we have the
Thus, in the MTP-model, we have the following two features of potential: a jump of the potential’s derivative on
surfaces of atomic spheres and a jump of the potential
magnitude on the surface of the molecular sphere. The
role of the potential jogs was also recently analyzed in [23]
in connection with the photoionization of so-called hollow
atoms (HA). It was shown that the amplitude of oscillations
in the energy dependence of the photoionization cross section of HA is extremely sensitive to the magnitude of the
potential derivative discontinuity. As far as the effect of
potential jumps on the particle wave function is concerned,
it has been thoroughly studied in connection to the square
well potential (see e.g. in [24, 25]).
It is easy to estimate the size of a potential jump if the
atomic ion V (r) potential is taken to be pure Coulomb.
In this case, the height of the potential barrier is ∆V =
1/Ra −1/Rm . For radii of spheres Ra ≈ 1 and also Rm ≈ 2,
∆V ∼ Ry. Thus, the height of the spherical potential
barrier surrounding a molecule in the MTP-model is not
small on atomic scales4 .
In the photoionization of a deep atomic level, the electronic wave that is distributed from the center of sphere I1
passes above this barrier. Its reflection leads to a change
in the amplitude of the wave function near the atomic
nucleus by F (k), and, hence, to a change in |F (k)|2 of
the photoionization cross-section (where k is the photoelectron’s linear momentum). Due to the connection between electron wave-function oscillations in the potential
The atomic system of units is used in this paper.
It is obvious that, with the inclusion −1/Rm , the polarization potential will not qualitatively alter the effects of
“jumps” in potential V (r) magnitude and its first derivative
in r.
4
Note that the potential well depth for C60 is two times
smaller, ∆V ≈ 8 eV [16, 17]. However, the presence of
this potential resonator radically alters the frequency dependence of the photoionization cross-section of an atom
located inside C60 in A@C60 .
2
3
827
Unauthenticated
Download Date | 6/18/17 10:38 AM
Non-physical consequences of the muffin-tin-type intra-molecular potential
resonator and outside of it, the amplitude F (k), a function of momentum k, looks like a fading sinusoid [26]. At
k 2 /2 ∆V , the potential jump ∆V is insignificant and,
therefore, F (k → ∞) → 1. The oscillating behavior F (k)
is already obvious based on general arguments, but the
magnitude of F (k) can be established only by numerical calculation. Note that oscillations F (k) considered in
spherically-symmetric systems with similar potential barriers are able to change the photoionization cross-section
of an atomic deep level near the threshold by several
times.
The diffraction effects in molecular photoionization are
approximately described by the function Fdif (k) = (1 +
sin kR/kR) that multiplies the photoionization crosssection of isolated atom ionization [27], where R is the
inter-atomic distance in a diatomic molecule. At the
threshold (k = 0), this function increases the photoionization cross-section by a factor two. Hence, the “handmade” barrier effects in the MTP-model are comparable
or even surpass the real diffraction ones and, therefore,
the research on those MTP effects with the aim of their
elimination is essential.
4. Spherically-symmetric potential
barrier
In order to analyze the role of discontinuities in the magnitude and first derivative of the potential, it is not necessary
to solve the Schrodinger equation with the MTP depicted
in Fig. 1. Indeed, in order to clarify the source of additional, as compared to the target’s, diatomic structure, oscillations in the photoionization come from discontinuities
of the MFT potential. For demonstration, we solve a simple model problem, allowing for analysis of the influence
of a potential step in the otherwise continuous spectrum
wave functions. Let us calculate the wave functions of a
hydrogen-like atom with nuclear charge Z, surrounded by
a potential barrier. It is obvious that, as a result of the
spherical symmetry of the atom-like system under consideration, this calculation reduces to obtaining the radial
part of the electron wave function Pkl (r) with linear momentum k in the continuous spectrum and with angular
moment l. The potential in which the electron moves is
defined as follows:
where atomic I1 and the molecular spheres touch each
other. From potential (1), it is obvious that wave function
Pkl (r) is a combination of Coulomb functions and spherical Bessel functions “sewed” together at points Ra and
Rm . This choice is determined by a desire to derive solutions analytically. Instead of −Z /r, more sophisticated
potentials could be employed. However, our aim here is
to study the role of “jumps” in the magnitude and first
derivative of the potential upon the wave function.
Let ukl (r) and υkl (r) denote regular and irregular, at r = 0,
Coulomb wave functions with asymptotic behavior:
πl
Z
+ δl ,
ukl (r) ≈ sin kr + ln 2kr −
k
2
πl
Z
+ δl ,
υkl (r) ≈ − cos kr + ln 2kr −
k
2
where δl (k) = arg Γ(l + 1 − iη) is the Coulomb phase shift.
Then, in the first region of r, the radial part of the wave
function is
Pκl (r) = Fl (k)uκl (r)
(r 6 Ra ).
V (r) = −Z /Ra − ∆V
V (r) = −Z /r
(r 6 Ra ),
(Ra 6 r 6 Rm ),
(1)
(r > Rm ).
Here, ∆V is the height of the potential jump at the point
(3)
The wave function in this area differs from the regular
solution of the Schrödinger equation with potential (1)
ukl (r) by only an amplitude multiplier Fl (k). The electron
wave vector κ in this area is defined by κ 2 /2 = k 2 /2 + ∆V .
In the second area of r,
Pkl (r) = Ajl (qr) − Bnl (qr)
(Ra 6 r 6 Rm ).
(4)
Here, jl (qr) and nl (qr) are the spherical Bessel functions [28] multiplied by qr, with asymptotic behavior
πl
,
jl (qr) ≈ sin qr −
2
πl
nl (qr) ≈ − cos qr −
.
2
(5)
The electron wave vector q in functions (4) and (5) is defined by q2 /2 = k 2 /2 + Z /Ra + ∆V .
In the third area of distances,
Pkl (r) = ukl (r) cos ∆l − υkl (r) sin ∆l
V (r) = −Z /r − ∆V
(2)
(r > Rm ).
(6)
Here, ∆l is an additional phase shift, acquired by the
electron wave function while passing the potential step.
At points r = Ra and r = Rm , the functions and their
derivatives should be sewed. This leads to the following
system of four equations:
828
Unauthenticated
Download Date | 6/18/17 10:38 AM
Miron Ya. Amusia, Arkadiy S. Baltenkov
∆l and the amplitude Fl according to the following equations:
Fl ukl (Ra ) = Ajl (qRa ) − Bnl (qRa ),
Fl u0kl (Ra ) = Ajl0 (qRa ) − Bn0l (qRa ),
tan ∆l (k) =
Ajl (qRm ) − Bnl (qRm ) = ukl (Rm ) cos ∆l
− υkl (Rm ) sin ∆l ,
Fl (k) =
0
− υkl
(Rm ) sin ∆l .
The prime mark (‘) denotes differentiation on r. Here,
unknown are coefficients A, B, Fl , and ∆l . All are functions
of k and l.
To find Fl and ∆l , we multiply the first equation by
n0l (qRa ), and the second by nl (qRa ), then deduct the first
equation from the second. The third equation is multiplied by u0kl (Rm ), and the fourth by ukl (Rm ), and deduct
them from each other. Finally, we obtain
W1 W4 − W2 W3
W1 W6 − W2 W5
.
−
W3 =
ukl (Rm )jl0 (qRm )
− u0kl (Rm )jl (qRm ),
W4 =
ukl (Rm )n0l (qRm )
u0κl (Ra )nl (qRa ),
− u0kl (Rm )nl (qRm ),
(9)
0
W5 = υkl (Rm )jl0 (qRm ) − υkl
(Rm )jl (qRm ),
0
W6 = υkl (Rm )n0l (qRm ) − υkl
(Rm )nl (qRm ).
For amplitude Fl (k), the following two equivalent equations are obtained:
qk sin ∆l (k)
,
W1 W4 − W2 W3
qk cos ∆l (k)
Fl (k) =
.
W1 W6 − W2 W5
(12)
Here, functions W7 and W8 are given by equations
W7 = uκl (Ra )u0kl (Ra ) − u0κl (Ra )ukl (Ra ),
0
W8 = uκl (Ra )υkl
(Ra ) − u0κl (Ra )υkl (Ra ).
(13)
We apply the obtained formulas in numerical calculations
of amplitudes Fl (k) for various heights and widths of the
potential step.
5. Dependence of the amplitude
Fl (k) on the height of the barrier
W1 = uκl (Ra )jl0 (qRa ) − u0κl (Ra )jl (qRa ),
W2 =
cos ∆l (k)
[ukl (Ra ) − υkl (Ra ) tan ∆l (k)] .
uκl (Ra )
(8)
Here,
uκl (Ra )n0l (qRa )
(11)
(7)
Ajl0 (qRm ) − Bn0l (qRm ) = u0kl (Rm ) cos ∆l
tan ∆l (k) =
W7
,
W8
Fl (k) =
(10)
Note that amplitude Fl (k) is always positive since, in this
case, the continuum wave function near zero in Eq. (3)
behaves as if in a centrally symmetric field, that is, as
Pκl (r → 0) ∼ r l+1 .
The case of a zero-width step (Ra = Rm ) requires special consideration. In this case, Fl and ∆l are determined
by equating the logarithmic derivative of Coulomb function (3) and (6) at this point. Substituting in (7) Ra = Rm
and equating the left sides of the first two equations and
the right sides of the second two equations, and repeating
calculations similar to the previous ones, yield the phase
Let us consider again the case of a zero-width potential
step, the case Ra = Rm = 1. Results of calculations of
amplitude F0 (k) with orbital momentum l = 0 using formulas (12) and (13) at various heights of barrier ∆V = 0.01,
0.1, and 0.5 a.u. are shown in Fig. 2. In this figure, the potential of the system V (r) under considerationis depicted,
formed by two Coulomb potentials
V (r) = −Z /r − ∆V
V (r) = −Z /r
(r 6 Ra ),
(r > Ra ).
(14)
In the calculations below, we take Z = 1. We take as
parameters the size of ∆V and the width of the barrier.
Functions F0 (k) in Fig. 2 look like periodically changing
functions with almost identical period ∆k ≈ 3.6. Deviations from perfect periodic behavior are observed only at
small electron kinetic energies. The amplitudes of the oscillations decrease with the growth of electron momentum
and reaches its asymptotic value F0 = 1. With an increase
in barrier height ∆V ,the amplitude of the oscillations, as
one would expect, grows.
In Fig. 3, the amplitude factor F1 (k) for the wave function with orbital moment l = 1 is represented. Here,
V (r) together with (14) includes the centrifugal potential
l(l + 1)/2r 2 = 1/r 2 for l = 1. This potential supersedes
the wave function from the area where there is a jump in
potential V . Therefore, at small ∆V = 0.01, the presence
829
Unauthenticated
Download Date | 6/18/17 10:38 AM
Non-physical consequences of the muffin-tin-type intra-molecular potential
Figure 2.
The amplitude factor F0 (k) as a function of the photoelectron’s linear momentum, s-wave for three values of the potential step ∆V .
Figure 3.
The amplitude factor F1 (k) as a function of the photoelectron’s linear momentum, p-wave for three values of the potential step ∆V .
of a barrier is insignificant and F1 (k) ≈ 1 in all the intervals of electron momentum under consideration. As in
Fig. 2, the amplitude’s factor oscillates with practically the
same period, ∆k ≈ 3.3. The difference is that all curves in
Fig. 2 change similarly, whereas the curves corresponding
to ∆V =0.1 and 0.5 at k ≈ 7.4 are strictly in anti-phase.
6. Dependence of Fl (k) on parameters of a potential step
The potential (14) corresponds, in the MTP-model, to the
case when an electron moves in a continuous spectrum
along trajectory 1, a trajectory for which the width of a
potential step is zero. It is clear that, in a molecule described by the MTP-model, a spherical electronic wave,
Figure 4.
The amplitude factor F0 (k) as a function of the photoelectron’s linear momentum, s-wave for four values of the potential step ∆V width w.
Figure 5.
The amplitude factor F1 (k) as a function of the photoelectron’s linear momentum, p-wave for four values of the potential step ∆V width w.
inasmuch as it is distributed in all directions from the center of atomic sphere I1 , experiences reflection from a potential step of variable width. It corresponds to an atom-like
system with potential (1). For numerical calculations, we
take the jump of potential on the border of the molecular sphere ∆V so that the average height of the potential
barrier on all possible trajectories of electron movement
is ∆V̄ = 0.5. We define it as the arithmetic average of
the minimal and maximal heights of a potential step. For
fixed atomic radius Ra = 1, the radius of the molecular
sphere is three times more, Rm = 3. Therefore, in order to
fulfill the condition ∆V̄ =0.5, it is necessary to substitute
into (1) the potential ∆V = 1/6.
The results of Fl (k) calculation at two different electron
orbital moments are represented in Figs. 4 and 5. In these
830
Unauthenticated
Download Date | 6/18/17 10:38 AM
Miron Ya. Amusia, Arkadiy S. Baltenkov
figures, the potential V (r), calculated according to (1), is
represented. The widths of potential step w = Rm − Ra =
0.5, 1.0, 1.5, and 2.0 were considered. Comparison of the
curves in Figs. 2 and 4 shows that the amplitudes F0 (k)
behave similarly. In both cases, they are fading, oscillating functions. But the oscillating curves in Fig. 4, like the
curves in Fig. 2, have different periods. This results from
the change in the potential radius Rm . Apart from the final width of a potential step, to be exact, the presence of
both types of features of potentialjumps in the derivative
dV /dr and in the potential V (r) modifies the form of the
curves, making them smoother.
7. Comparison with a smooth potential change
In order to confirm that the observed peculiarities depicted in Figs. 2-5 are non-physical and result from a
“jump” in (14) at r = Ra , we calculated wave functions
with smooth additional potential Vadd (r) instead of (14).
It follows that the equation for Pκl (r) from (3) is given by
Z
∆V
1 d2
− +
−
2
2 dr 2
r
1 + e(r2 −Ra )/α 2
l(l + 1)
(α)
+
E
Pkl kl (r) = 0.
−
2r 2
Acknowledgements
This work was supported by the Hebrew University intramural fund and Uzbek Foundation Award ΦA-Φ4-Φ100.
(15)
Like Vadd (r) in (15), we use potential with square dependence upon r instead of the more familiar nuclear shell
modelthe so-called Woods-Saxon potential Vadd (r) =
∆V /[1 + exp β(r − Ra )] because the latter is non-analytic
as a function of ~r at ~r = 0. Using the solution of (15), one
(a)
(a)
can introduce Fkl = Pkl (r)/ukl (r)|r→0 . It is clear that
(∞)
Fkl = 1. As α → 0, the potential in (14) coincides
(0)
(a)
with (14), and Fkl → Fkl ≡ Fkl . Concrete calculations
were performed for an s -wave with Ra = 1, l = 0, and
(a)
∆V = 0.5. The factor Fk0 , as a function of α, was found
to deviate from Fk0 very rapidly, already approaching the
value 1 when α > 0.2 for any k. This demonstrates the
observation that only a sharp increase in the atomic scale
can mimic a “jump.” To be physical for a molecule, this
variation has to be much smaller than the atomic radius
that is confirmed by the obtained number of limiting α.
8.
electron energies by more than an order of magnitude, an
amount considerably stronger than the variations due to
the diffraction factor Fdif (k) = (1 + sin 2kRa /2kRa ) that
correspond to a molecule formed by two identical atoms
(Fig. 1). For such molecules, the inter-atomic distance is
2Ra . The diffraction period is close to ∆k ≈ 3.14, the
period of functions F0 (k) and F1 (k). This means that the
barrier and diffraction effects have similar periodic structure. However, the maximal squares of barrier functions
essentially surpass Fdif (k).
Let us conclude by giving another example of the consequences of the non-analytic behavior of potential. It is
worth noting that the non-analyticity of potential V (r) =
V0 exp(−ar) as a function of ~r leads to decreased power
in the photoionization cross section in the high photon
energy ω -limit as a function of ω, while the analytic
potential V (r) = V0 exp(−br 2 ) leads to an exponentially
decreasing cross section despite the fact that, at small
distances corresponding to high momentum and energy,
the potentials are equal.
Conclusions and discussion
As mentioned above, the photoionization cross-section
is proportional to the square of Fl (k). Therefore, nonphysical barrier effects in the model (14) under consideration change the photoionization cross-section at resonant
References
[1] F. Herman, S. Skillman, Atomic Structure Calculations (Prentice–Hall, Inc., Englewood Cliffs, NJ, 1963)
[2] D. Dill, J. L. Dehmer, J. Chem. Phys. 61, 692 (1974)
[3] L. Vitos, H. L. Skriver, B. Johansson, J. Kollár, Comp.
Mater. Sci. 18, 24 (2000)
[4] Shih-I Chu, J. Chem. Phys. 123, 062207 (2005)
[5] M. Ya. Amusia, I. M. Band, V. K. Ivanov, V. A.
Kupchenko, M. V. Trzhaskovskaya, Izv. Akad. Nauk
SSSR Fiz.+ 50, 1267 (1986)
[6] Y. Kuang et al., J. Phys.-Paris 48, C9-527 (1987)
[7] Zhou Bin, R. H. Pratt, Phys. Rev. A 45, 6318 (1992)
[8] M. Hansen, H. Nishioka, Z. Phys. D 28, 73 (1993)
[9] M. E. Madjet, Himadri S. Chakraborty, Jan–M. Rost,
J. Phys. B-At. Mol. Opt. 34, L345-L351 (2001)
[10] T. L. Loucks, The Augmented Plane Wave Method
(Benjamin, New York, 1967)
[11] D. Dill, J. Siegel, J. L. Dehmer, J. Chem. Phys. 65,
3158 (1976)
[12] C. T. Chen, Y. Ma, F. Sette, Phys. Rev. A 40, 6737
(1989)
[13] D. L. Lynch, Phys. Rev. A 43, 5176 (1991)
[14] F. Heiser et al., Phys. Rev. Lett. 79, 2435 (1997)
831
Unauthenticated
Download Date | 6/18/17 10:38 AM
Non-physical consequences of the muffin-tin-type intra-molecular potential
[15] E. Shigemasa et al., Phys. Rev. Lett. 80, 1622 (1998)
[16] M. J. Puska, R. M. Nieminen, Phys. Rev. A 47,
1181(1993)
[17] O. Frank, J.-M. Rost, Chem. Phys. Lett. 271, 367 (1997)
[18] M. Ya. Amusia, A. S. Baltenkov, V. K. Dolmatov, S. T.
Manson, A. Z. Msezane, Phys. Rev. 70, 023201 (2004)
[19] L. V. Azaroff, Rev. Mod. Phys. 35, 1012 (1963)
[20] E. A. Stern, Phys. Rev. B 10, 3027 (1974)
[21] R. de L. Kronig, Z. Phys. 70, 317 (1931)
[22] R. de L. Kronig, Z. Phys. 75, 191 (1932)
[23] A. S. Baltenkov, S. T. Manson, A. Z. Msezane, Phys.
Rev. 76, 042707 (2007)
[24] L. I. Schiff, Quantum Mechanics (New York, McGrawHill, 1949)
[25] L. D. Landau, E. M. Lifshitz, Quantum Mechanics.
Non-Relativistic Theory (Pergamon, Oxford, 1965)
[26] M. Ya. Amusia, A. S. Baltenkov, U. Becker, Phys. Rev.
A 62, 012701 (2000)
[27] H. D. Cohen, U. Fano, Phys. Rev. 150, 30 (1966)
[28] M. Abramowitz, I. A. Stegun (Eds.), Handbook of
Mathematical Functions (Dover, New York, 1965)
832
Unauthenticated
Download Date | 6/18/17 10:38 AM