CC BY
26
2014
ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА
Сер. 4. Том 1 (59). Вып. 2
ФИЗИКА
УДК 530.146.6 I. Yu. Yurova
EFFECTIVE ¿-DEPENDENT POTENTIAL FOR THE SYSTEM: ELECTRON — POSITIVE ION OF GOLD ATOM. ORBITAL ENERGIES
St. Petersburg State University, 199034, St. Petersburg, Russian Federation
We present relativistic effective ¿-dependent potentials (RELP) for electron — ion Au+. We determined parameters of ELP from Auger and optical spectroscopic data. Orbital energies of atom we calculated by two approaches: non-relativistic with Schrodinger equation and relativistic one with Dirac equation. We calculated bound energies of 34 orbital states of Au atom with presently developed effective potential RELP. Results are in good agreement with Dirac—Hartree—Fock calculations and experimental data. We applied potential RELP for oscillator strengths calculations. Our results of oscillator strengths values agree with Dirac—Hartree—Fock these. With the combination of RELP and core polarized dipole correction method we received oscillator strengths conforming to experimental data. We applied the RELP to calculation of the electron-electron screening constant x in metallic Gold. For these purposes, we calculated the 4/7/2 orbital energy shift of Gold in metallic phase applying perturbation theory with RLEP. From the comparison of calculated and experimental values of 4/7/2 energy shift we estimated the value of electron-electron screening constant as 0.08 < X < 0.3 for different systems formed by solid gold. Refs 22. Figs 1. Tables 5.
Keywords: effective potential, ion, atom, spectroscopy, Gold, orbital energy, calculation, oscillator strength, energy shift, fine structure.
1. Introduction. Considerable efforts are directed to experimental and theoretical studies of Gold atom and Gold systems such as semi-infinite media, thin films, quantum wires, nanoparticles for many years until nowadays. One can get knowledge about Gold and Gold systems from the special edition: Gold Bulletin (World Gold Council, London). There are basic experimental methods in the field of the examination of Gold: spectroscopic determination of high energy levels of Au atom [1, 2], X-ray photoemission spectroscopy [3] and Auger spectroscopy [4] for core levels. Above spectroscopic methods, there is another type of experiment — the elastic peak electron spectroscopy (EPES) [5]. EPES deals with the electron scattering from the solid Gold and Gold system surfaces at intermediate and high non-relativistic impact energies.
Theoretical investigation of heavy atoms electronic structure deals with the Dirac—Hartree—Fock (DHF) [6, 7] calculations. Either all electrons of an atom or valence electrons only are taken into consideration in self-consistent field calculations. In latter case, all electrons except valence these, are substituted by effective potential [8]. Note, that DHF results are not always accurate, especially for high energy levels (see Table 1). There are already known
electron—Gold effective potentials [9, 10] however, these potentials did not provide proper values of Au orbital energies.
Table 1
Au orbital energies (all are negative), and radial expectations
Level Orbital energy Rni
Present DHFS IPM DHF Experiment Present DHF
7s 0.0899 - 0.0849 0.0847 0.0908 [1] 8.61 9.09
0.0897rei 8.67rei
6 s 0.3390 0.1376 0.2861 0.2919 0.3390 [12] 2.82 3.06
0.3367rei 2.83rei
5s 3.940 1.063 3.671 4.688 3.934 [2] 0.94 0.84
3.827rei 0.94rei
4s 28.01 17.23 23.67 29.15 28.01 [3] 0.41 0.41
27.93rei 0.41rei
3s 126.0 91.95 106.5 128.1 126.1 [3] 0.20 0.18
130.5rei 0.18rei
■2s 544.7 406.0 454.7 532.4 527.6 [3] 0.079 0.070
540.4rei 0.070rei
Is 2855 2592 2712 2988 2967 [3] 0.019 0.017
299 Sj'gj 0.017rei
7p 0.0612 - 0.0578 0.0555 0.0630 [1] 11.6 12.6
0.0602rei 11.5rei
6 p 0.1601 0.0842 0.1401 0.1317 0.1570 [1] 4.12 4.88
0.1556rei 4.10rei
5 p 2.412 0.508 2.507 2.770 2.310 [3] 1.07 1.06
2.276rei 1.07rei
4p 21.87 13.59 20.21 22.29 21.74 [3] 0.43 0.45
21.17rei 0.44rei
3p 108.3 82.50 98.60 107.7 106.0 [3] 0.19 0.18
107.7rei 0.20rei
■2p 455.1 381.7 439.7 464.2 460.5 [3] 0.068 0.062
459.1rei 0.063rei
5 d. 0.4336 0.3439 0.6297 0.4546 0.4284 [2] 1.76 1.54
0.4584rei 1.84rei
4d 11.33 7.326 13.61 13.45 12.72 [2] 0.46 0.46
11.78rei 0.46rei
3d 82.32 65.85 82.95 84.02 82.47 [2] 0.17 0.17
83.92rei 0.17rei
4/ 3.155 0.0407 4.660 3.784 3.146 [3] 0.47 0.50
3.117rei 0.44rei
All values are in atomic units. Present unmarked — nonrelativistic calculations; "rel" — relativistic ones. Relativistic values with l > 0 were averaged over (l — 1/2, l + 1/2) configuration.
The goal of the present consideration is to develop an approach as to Au orbital states and to electron—Au scattering calculations as well to obtain theoretical results corresponded to above referred spectroscopic and EPES experiments. For these purposes we developed the e-—Au+ (V(+)) /-dependent non-relativistic and relativistic effective interaction potentials and considered its applications to calculations of orbital energies, electron densities, dipole oscillator strengths of Au atom and orbital energy shift calculation.
2. Effective electron — ion Au+ potential (ELP).
2.1. Analytic effective potentials for the electron — heavy ion interaction. In
spite of great variety of effective potentials, only few of them have been developed for the modeling the interaction between an electron and heavy atom with nuclear charge Znuc > 70. For example, it is the Independent Particle Model (IPM) effective potential:
^nuc-n |
ViPM(r) = --^-, Q(r) = H(exp(r/d) - 1) + 1, (1)
r
where n = 1 for electron — one charged positive ion interaction; n = 0 for electron — neutral atom interaction; H = d(Znuc — n)0'4, the parameter d is tabulated for Znuc = 2 + 103; = VipmI,1=i and V^jp^ = VipmI,1=o- F°r Au and Au+ the parameter d = 0.657.
The second effective potential is the Dirac—Hartree—Fock—Slater (DHFS) one [10]:
Znuc 3
VdhfsM =--— y^ Ai exp(-arr), (2)
i=i
y(+) - i i
VDHFS S 1
vdhfs, r < ro ;
r > ro,
vdhfs at 0 ^ r < <x. The values of parameters in the potential (2) are tabulated for Znuc = 2 ^ 91. For Au and Au+ ion they are equal to: A1 = 0.2289, A2 = 0.6114, A3 = 1 — Ai — A2, ai = 22 864, a2 = 3.6914, a3 = 1.4886.
We calculated orbital energies of Au atom with potentials vipm and vD+Fs, Table 1. One could conclude, that those potentials especially, did not provide accurate values of orbital energies (see Table 1). That is why we developed new e-—Au+ effective potential provided proper values of orbital energies. Moreover, it occurred, that our e- — neutral atom Au effective potential provided accurate elastic e-—Au+ cross sections (see below).
2.2. Electron—Au+ Z-dependent effective potentials. According to the theory of /-dependent effective potentials [8, 16], we used the expression:
V (+)(r) = £ Vl{+)(r)\l)(l\, (3)
i
where \l)(l\ is the projection operator in Hilbert space onto the subspace with fixed orbital momentum l. The exchange and Coulomb potentials are usually included in the V(+) (r) [8, 11]. In present consideration we did not include the core polarization, exchange and correlation in the V(+)(r) separately. However, as indicated in Table 1, the successful fit to orbital energy values suggests that the present effective potential includes effects of exchange and correlation. Radial potentials V^+^r) in (3) are determined as:
ii
^(+)(r) = 14 + ]T Cj(+V exp (-<zf V) - i, (4)
i= — 1
exp (-q(_\A
Vh=Cl_ 2 (5)
r(r + h)
From the expansion of V^+^r) at small r, one can obtain the equality:
Ci(+)
+ Cl-V - 1 = "^nuc (6)
r
We assume that at l > 2 potentials coincide with the potential V^. Parameters of (4), (5) we find from the fitting of calculated bound atom energies to experimental data analogized [9, 11]. Note, that we could not reach the success in fitting until the additional term Vh (5) was included in expression (4). The fitting way allows anybody to use the spectroscopic data bank with accurate values of atom bound energies known for the most of atoms of periodic table. To calculate orbital energies we solved numerically radial Schrodinger and Dirac equations with potentials (4), (5) and obtained parameters of effective l-dependent non-relativistic potential (ELP) and relativistic one (RELP) correspondingly. For fitting we have applied reference states 4s, 5s, 6s, 4pj, 5pj, 6pj and 4dj, 5dj, 4fj, with averaging over j for ELP, and with one value of j from two possible these (j = l ± 1/2) for RELP; experimental data had been taken from [1-4, 12]. Note, that the parameter one can find from the condition (6). For calculations we applied our own computer code in non-relativistic case, for Dirac equation we applied the DHF computer code presented by professor 1.1. Tupitsyn. In Table 2 parameters of ELP and RELP radial potentials (4), (5) are presented. Both VELP and Vrelp potentials are given by same expressions (3)-(6) with same values of all parameters except the parameter h.
Table 2
Parameters of the potential (2)
i (0 Ii
I = 0 h = 0.100 h = 0.068 (rel) -2 2.174 -1.182
-1 3.020 -66.180
0 3.429 26.450
1 2.224 -21.074
I = 1 h = 0.072 h = 0.0615 (rel) -2 2.310 -1.211
-1 2.890 -60.764
0 2.9000 19.700
1 2.135 -20.700
I > 1 h = 0.0304 h = 0.02954 (rel) -2 2.300 -0.438
-1 2.800 -63.400
0 2.800 37.020
1 2.030 -22.970
All values are in atomic units.
We present results for orbital energies and radial expectations in Table 1. We inserted in Table 1 values, obtained with DHS all electrons method, and with effective potentials (1) and (2) as well. There is good agreement of present orbital energies with experimental data and DHF results in both non-relativistic ELP and relativistic RELP calculations. We present deviations in percents of calculated orbital energies from experimental values in Table 3. ELP and RELP 3s, 3p, 7s, 6p, 7p energies agree experimental data and DHF results without fitting. Note, that RELP 1s, 2s, 2p energies agree DHF and experimental data better than non-relativistic ones. The potential vD+Fs appeared poor, and the potential VIPM seems slightly better for calculations of orbital energies of Au atom (Table 3). Energies of spinorbit splitting of Au for n = 2 + 5 are presented in Table 4. The accuracy of RELP fine structure results is about 0.5-10 %.
3. Dipole oscillator strengths. We employ radial potentials (2) in the calculations of Au dipole oscillator strength fYj,Y',j'. According to [13], it is the function of matrix elements from components of the electron dipole moment Di and energies of bound states Ej, EY'j'
Table 3
Deviations of calculated orbital energy from experimental values (in percents)
'2p-&p 3d-5d. 4/
Vipm 7-15 5-11 6-32 34
Vdhfs 13-59 17-78 20-46 87
I'present. 0.006-4 0.3-4 2-8 0.5
Relativistic values with l > 0 were averaged over (l — 1/2, l + 1/2) configuration.
Table 4
Relativistic nlj orbital energies (all are negative) and radial expectations
Level Orbital energy Rnij
RELP DHF Experiment RELP DHF
7p 0.0631/2 0.0581/2 0.065I/2 [1] ll.l1/2 12.2I/2
0.0593/2 0.0543/2 0.0623/2 I2.I3/2 I3.O3/2
6 p 0.1751/2 0.140I/2 0.169I/2 [1] 3.8O1/2 4.61I/2
0.1453/2 0.1273/2 O.I5I3/2 4.483/2 5.I63/2
5 p 2.651/2 3.193/2 2.73I/2 [1] 1.02I/2 1.02I/2
2.093/2 2.561/2 2.IO3/2 1.123/2 I.H3/2
4p 23.601/2 24.741/2 23.63I/2 [3] O.4O1/2 0.41I/2
19.913/2 21.073/a 2O.8O3/2 0.433/2 0.443/2
3p 117.81/2 117.8I/2 115.9I/2 [3] 0.17I/2 0.17I/2
102.73/2 102.73/2 IOI.O3/2 0.193/2 0.193/2
■2p 503.41/2 509.3I/2 504.9I/2 [3] O.O61/2 O.O61/2
437.03/2 441.73/2 438.33/2 0.073/2 0.073/2
5 d. 0.4803/2 0.4933/2 0.4593/2 [2] 1.693/2 1.543/2
0.444b/2 0.429b/2 O.4O85/2 1.74b/2 1.62b/2
4d I2.I83/2 13.873/2 13.123/2 [2] 0.443/2 0.453/2
11.51b/2 13.175/2 12.46b/2 0.45b/2 0.46b/2
3d 85.763/2 85.953/2 84.343/2 [2] O.I63/2 O.I63/2
82.89b/2 82.74b/2 81.225/2 0.17b/2 0.17b/2
4/ 3.225/2 3.87b/2 3.22b/2 [3] 0.46b/2 0.49b/2
3.047/2 3.727/2 3.097/2 0.477/2 0.507/2
All values are in atomic units.
/v:
2(EY'j' )
\Dyj,y',j' II ,
iyj,Y,f- 3(2j + 1) where y, y' are all quantum numbers except j, j' ones,
WDjfj'W2 = E E \(yJM\Di\y'J'M')\2,
i M,M'
index "i" designs x, y, z. For transitions 6s1/2 ^ npj, j = 1/2, 3/2, we have:
(7)
/ \ 2 / \
(P6s1/2 r Pnpj J + {QeSl/2 r Qnpj)
, a 1/2 = 2, a3/2 = 4, (8)
2
Pnli, Qnli are large and small components of Dirac wave functions. We calculated dipole oscillator strengths for 6s ^ npi/2 and 6s ^ np3/2 transitions for n = 6, 7 in RELP, we applied dipole correction method with core polarization, as it was done in [14], and in DHF approximations. We present results for dipole oscillator strengths in Table 5.
Table 5
Oscillator strengths of 6sj —> npj transitions of atom Au, n = 6, 7
Transition Present Present CPDC DHF* Other theories Experiment
6si/2 &P3/2 0.705 0.340 0.609 0.791** 0.623*** 0.36**** 0.08-0.41 [16] 0.344(16) [17]
6Si/2 &P1/2 0.319 0.172 0.285 0.390** 0.296*** 0.158**** 0.06-0.19 [16] 0.170(7) [17]
6Si/2 7p3/n 3.16(—2) 4.07(—3) 5.48(—2)
6Si/2 7p1/2 2.36(—3) 1.08(—3) 1.69(—2)
7*1/2 <'p3/2 0.912 0.879
7*1/2 7j5l/2 0.905 0.946
Present — relativistic effective ¿-dependent potential (RELP), CPDC — core polarization dipole corrected method.
* Energy values and wave functions were calculated by professor I. I.Tupitsyn.
** DHF [15].
*** Many configuration DHF [15].
**** CPDC DHF [14].
One can see in Table 5, that DHF method [15] and presently applied DHF with RELP approximation give oscillator strengths approximately twice higher experimental ones, however our dc calculations and the theory [14] gives values being in the good agreement with experimental data [16, 17]. One could explain this agreement by the inclusion in DHF the CPDC effect opposite other calculations.
4. Orbital energy shifts in solid gold and estimation of electron—electron screening constant. The other application of RELP is the calculation of orbital energy shifts in solids. 4f orbital of Au atom is the most suitable one for the investigation orbital energy shift due to various perturbation effects in solid state [4]. One can write the electron—electron interaction potential in solids in accordance with [18, formula (8)]:
Vee(r, r') = C
exp( — |r — r' |/X)
(9)
C = 0 for free atom and C = ±1 for bulk metals, clusters with different impurities and nanoparticles with different sizes and pretreatment conditions. We consider Vee as the perturbation causes of atom orbital energy shift. In order to calculate the correspondent energy shift we averaged the interaction Vee(r, r') over Wigner—Zeitz cell with plane wave approximation for Bloch functions [19]:
Vee(r,r') = —!— f Vee(r,r')dr' ^LWZ J
WZ
(10)
where ilWZ — the volume of Wigner—Zeitz cell; R0 = 2.99ao for Gold [20]. In order to calculate integral in expression (10) we applied the expansion of Vee(r, r') onto a sequence
rr
84.0
£
M 83.5
83.0
84.22 -
84.18 -
84.14 -
84.1
0.1 0.15 0.2 0.25 0.3 l, A
0.02 0.04 0.06 0.08 0.1 0.12
l, A
The dependence of 4f7/2 orbital energy on electron—electron interaction constant X
in metallic gold system:
a — C = +1; b — C = -1
of modified Bessel functions with variables r/X,r'/X (see [18, formula (13)]). In result, we obtained the expression for averaged electron—electron interaction potential:
Vee(r, r')
c-
Ro
X
exP(-;--exPl--
r < R0
(11)
.0, r> Ro.
In the first order of perturbation, the energy shift is equal to the sum of matrix elements:
Df
3±1/2
P4f Vee
4f3±1/2 ee
Pf
3±1/2
+ (Q4/.
3±1/2
Ve
Q4f.
3±1/2
(12)
where P4/3±1/2 and Q4/3±1/2 are large and small components of RELP orbital wave function 4f3±1j2. Applying expression (11), we obtained the 4f7/2 orbital energy as the function of electron-electron screening constant X for C = ±1, see Figure. One can apply the experimental values of 4f7/2 binding energy shift in number of nanoparticles produced by different pretreatments [21] and alloyed systems with solid Au [22] for estimation the electron—electron screening constant values; in result we obtained the interval 0.08 < X < 0.3.
5. Conclusion. We presented new relativistic effective l-dependent potential RELP for electron—Gold interaction. It was been successfully applied to the Gold atom orbital energies and electron densities calculations opposite to other known effective potentials.
We applied RELP to the calculation of dipole oscillator strengths of Au atom. The inclusion in RELP of dipole correction due to the core polarization effect allowed us to get results well agreed with experimental data.
The new way of estimations of electron—electron screening interaction constant X in metalls due to the calculation of orbital energy shift with RELP and fit theoretical results to experimental data was been proposed. Calculations of X were made for different systems formed by metallic Gold.
Aknowlegements. The author is pleased to acknowledge Professor I.I. Tupitsyn for giving an opportunity for application of the computer code for the solution of Dirac equation with effective potential. The author is thankful to Professor I.I. Tupitsyn for the presentation of all electrons DHF orbital energies, wave functions and radial expectations of Au atom.
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Статья поступила в редакцию 20 ноября 2013 г.
Контактная информация
Yurova Inna Yurievna — Dr. Sci., Professor; e-mail: [email protected]
