J. Phys. Chem. 1995,99, 2127-2133
2127
A Model for Electron-Zeolite Na+-Zeolite Interactions: Frame Charges and Ionic Sizes N. P. Blake, V. I. Srdanov, G. D. Stucky, and H. Metiu* Department of Chemistry, University of California, Santa Barbara, California 93106-9510 Received: September 7, 1994; In Final Form: December 2, 1994@
We construct and test empirical potentials for the interaction of the sodalite framework with an electron injected inside the zeolite and with the alkali counterions present in the cage. The electron interacts with Gaussian charge densities centered at the position of each zeolite atom and characterized by a net charge and a size. The magnitude of the charge is provided by ab initio calculations or by electronegativity arguments and the size by the crystal ionic radius. The ion-frame interaction contains a short-range Born-Mayer repulsion, as well as a dispersion and an induced dipole-monopole interaction. In this model the Coulombic interaction between the counterion and the frame atom uses the same frame charges as the ones used in the electron-frame interaction, and not the formal charges, commonly used in many ion-frame potentials. The effect of the long-range Coulombic interaction between the counterions and the counterions and the frame is calculated by using the Ewald summation. The electron-frame Coulombic interaction is calculated by using a fast Fourier transform method to solve Poisson’s equation. These potentials are tested by calculating the absorption spectrum of an electron “solvated’ in dry sodalite and in halosodalites. The computed peak frequencies agree well with those measured. In particular, we reproduce the linear dependence of the maximumabsorption wavelength on the lattice constant.
I. Introduction The properties of a molecular absorbed in a zeolite or a molecular sieve are different from those of its gas-phase counterpart.’ In some cases, which are not considered here, these differences are caused by a chemical reaction with the zeolite. There are however a large number of systems in which no frame-guest chemistry takes place, and the properties of the guest molecule are modified by the electric field produced by the frame and the counterions. The disproportionation chemistry of NO in zeolites,2the penetration of NaCl into zeolite cases at temperatures below the melting point of NaC1,3 and the stabilization of charge transfer transition states in photooxidation of alkenes in the presence of 0 2 4 are a few examples of processes for which these electric fields are held responsible. The changes caused by the local electric fields in zeolite are similar to those observed when the molecule is dissolved in a polar solvent; for this reason we often say that the absorbed molecules are solvated by the zeolite. The properties of these solvated molecules can be calculated by adding, to the Hamiltonian commonly used in gas-phase quantum chemistry, the interaction between the electrons of the molecule, and the electric field produced by the counterions and the zeolite frame. Such calculations are of the same order of difficulty as those performed in gas-phase chemistry and are therefore possible for molecules of practical importance. The need for performing such calculations makes the determination of the electronzeolite interaction a very important task. In this article we present and test potentials describing the interaction of an electron with zeolites from the sodalite family and the interaction of an alkali counterion with the zeolite frame. We have chosen this family for several reasons. First, the structures of these compounds are ~ e l l - k n o w n , ~and - ~ since the AYSi ratio is 1, the location of the A1 and the Si atoms on the lattice sites is unambiguous. Second, the properties of electrons solvated in sodalite^,^^'^ including the absorption spectra1’-14
* Author for correspondence.
’Abstract published in Advance ACS Abstracts, February 1, 1995. 0022-365419512099-2127$09.00/0
needed for testing the potentials, have already been studied experimentally. The structure of the sodalites is obtained by placing cages, like the one shown in Figure 1, on a body-centered-cubic lattice. The unit cell of the crystal obtained in this way has two cages and has P43m symmetry. The A1 and Si atoms are located at the cage vertices and are joined by 0 atoms (not shown in Figure 1). The AUSi ratio is unity, and each vertex occupied by an A1 atom is neighbored by Si-occupied vertices (Lowenstein’s rules), and vice versa. Since the A1 atoms are forced to be tetravalent, there are three negative charges per cage. Overall neutrality is maintained by the presence of three Na+ ions inside each cage. The simplest compound from this class is the “dry” sodalite, which has the stoichiometric formula (Na+)3(AlSi04-)3. When exposed to alkali vapor, this material turns blue>J0J4 The color appears because the alkali atoms absorbed inside the sodalite are ionized. The electron produced by ionization is “shared’ by the Na+ ions present in the cage9,10,14 and absorbs visible light. The absorption spectrum depends strongly on the interaction of the electron with the sodium ions and the frame and also on the position of the sodium atoms in the cage. For this reason it can be used to test electron-frame and ion-frame potentials. The stoichiometric formula of the color center in the dry sodalite is (Na+)4(AlSi04-)3e, where e stands for the electron. There are other sodalites having the same color center. These can be prepared by exposing the halosodalites (Na+)4(AlSi04-)3X-, where X = C1, Br, and I, to alkali vapor,1° to X-rays, or to an electron beam.13 It is believed that the electron produced by these procedures migrates through the zeolite until it finds and joins sites where the X- ion is accidentally missing. After they trap the electron these sites have the same stoichiometric formula (i.e., (Na+)4(AlSi04-)3e) as the color centers obtained by exposing the dry sodalite to alkali vapor. There are, however, differences. In halosodalites the cage holding the solvated electron is surrounded by (Na+)4(AlSi04-)3Xcages, while the solvated electron in the dry sodalite is surrounded by (Na+)3(AlSi04-)3. We can test the quality of
0 1995 American Chemical Society
Blake et al.
2128 J. Phys. Chem., Vol. 99, No. 7, 1995
dry and halosodalites are of interest on their own. The calculations presented here give the electron wave function, determine the degree of localization of the electron, and explain the empirical correlation between the peak frequency in the absorption spectrum of the solvated electron and the lattice constant of the sodalite. We have also calculated the absorption line shape in terms of the counterion motion after photon absorption.2z This method for devising an electron-frame potential can be extended to other zeolites since many of them ionize alkali atoms.23 Even if ionization does not occur, the absorption spectrum of the alkali atoms will be shifted when the atom is absorbed in a zeolite, and this shift can be used as a probe of the electron-frame interactions.
11. The Model Figure 1. Structure of the ,&cage of sodalite. The vertices are occupied by either Si or A1 atoms. (Each Si atom has A1 atom neighbors, and vice versa.) The vertices are joined by 0 bridges (not shown).
the proposed potentials by calculating the absorption spectra of these compounds and comparing them with the experiments. To describe the electron-frame interaction, we give each frame atom a spherically symmetric Gaussian charge density. This is characterized by two parameters: a total atomic charge and a length. The electron-frame interaction is given by the Coulombic interaction with the sum of these charge densities. One difficulty in designing a good electron-frame interaction is the abundance of parameters suggested in the literature. 15-17 The charges used here are based on ab initio calculations1*or electronegativity arguments, l9 and the lengths are given by the crystal ionic radii.20 This choice is validated by the agreement between the calculated absorption spectra and the experiment. Haug et aLZ1have pointed out that the spectrum of the solvated electron is very sensitive to the positions of the Na+ ions. Therefore, to describe the spectrum correctly, we also need a good ion-frame interaction. A successful procedure for generating this interaction gives formal charges to the atoms on the frame.” Unfortunately, if formal charges are used for the electron-frame interaction, they lead to a poor description of the absorption spectrum.z1 Because of this we replace the formal charges in the frame-ion interaction with the charges that we use in the electron-frame potential. This defines the Coulombic interaction between the ion and the zeolite. To this we add several other terms: a short-range Born-Mayer repulsion between Na+ and the oxygen atoms, a dispersion energy, and a monopole-induced dipole energy. The last two are longer-ranged and are used for the interaction of the Na+ ion with each frame atom. The only adjustable parameter is the range of the repulsive Born-Mayer potential (i.e., the length scale appearing in the exponent). This is varied until the positions of the Nai ions in dry sodalite coincide with those obtained from X-ray diffraction.8 All other parameters are taken from the literature. The present work can be viewed as an improvement of that of Haug et aL21 Several refinements in computational methodology allowed us to find the minimum-energy positions of the Na+ ions interacting with the electron and to use a more extensive data set to test the potential. As a result, the potential proposed here is better than the one derived earlier.21 Moreover, the ion-frame potential proposed here is new; the previous workz1 did not need one since in those calculations the ions were held fixed. This paper focuses on the electron-frame and ion-frame interaction. However, the properties of a solvated electron in
1. Structural Information. The atomic positions for these materials are provided by Rietveld analysis of the pertinent X-ray diffraction measurements. For the dry sodalite we use the structure derived by Felsche et aL8and for the chlorosodalite that of Hassan and G r u n d ~ ,while ~ for the bromo- and iodosodalites we take the structures of Beagley et aL6 In all systems the color center is (Na+)4(AlSi04-)3e. 2. Electron-Frame Interaction. We follow Haug et aL21 and describe the electron-frame interaction by an empirical pseudopotential which assigns each atom a charge and a size. Similar models provide a useful description of the behavior of an electron solvated in a l i q ~ i dor~ a~ luster.^^,^^ ,~~ According to the previous calculations*l and to the ESR experiment^,^.'^ much of the density of an electron solvated in a sodalite is localized in the interior of the tetrahedron formed by the four Na+ ions. The probability that the electron gets close to the frame is low: the electron is repelled by the negative 0 atoms located on the frame and is attracted by the Na+ ions. Thus, the electron does not get close to the positive A1 and Si atoms. For this reason the electron’s behavior is dominated by the longrange Coulombic interaction with the charges on the frame atoms, and the importance of these charges varies with their distance from the electron charge density. Thus, a correct treatment of the electron interaction with Na+ is more important than that with 0, Si, and Al. This is fortunate since, while there is wide agreement regarding the Na+-electron interaction, there is little consensus regarding the others. Based on these physical arguments, the interaction of the electron with the sodalite frame and the counterions is described by giving each atom i a Gaussian charge density centered at the atomic position Ri, having a width Ai and a total charge qi: ei(R) = qiAi-3(2n)-3/2 exp[-(IR, - R1/21/zAi)zl
(2.1)
This charge distribution generates the following potentialz8 (given here in atomic units): Vi(lRi - RI) = -qi erf(lRi - R1/2/2Ai)/lRi - RI
(2.2)
This replaces the Shaw p~tential:~,~~ used in the previous work.*l Both potentials have a Coulombic form at large distances. The Shaw form truncates the Coulombic interaction at short distance and replaces it with a constant. The cusp introduced by this truncation causes severe numerical inefficiency when fast Fourier transform methods are used. Equation 2.2 removes this deficiency (see section IV). The charges qi (see Table 1) are taken from STO-3G calculation^^^ performed on small model clusters. These are fairly close to the values of
Electron-Zeolite and Na+-Zeolite Interactions
J. Phys. Chem., Vol. 99, No. 7, 1995 2129
TABLE 1: Parameters Used in the Electron-Frame Interactiona atom Na Si A1 0
atom
Qi
Ai
+1.0 +1.0
1.16
c1
0.54 0.53 0.57
Br
-1.0 -1.0
I
-1.0
~
+0.8
-0.7
Ai
ai ~~
~~~
1.67 1.82 2.06
q, is the charge on framework atom i (in units of electronic charge), and Ai is the width of the Gaussian charge distribution for atom i (in angstroms).
-0.642e for 0, 0.96e for Si, and 0.61e for Al, inferred from the electronegativity method of Skorczyk.19 The oxygen charge recommended by Sauer18 is close to that obtained by Van Genechten et who use ab initio methods along with electronegativity equalization methods to assign partial charges for silica sodalite. For Na+, Si, and Al, the width Ai is taken equal to the ionic crystal radius for atom i with the coordination number given by Shannon.20 Since we do not have data for the coordination number of the halogens in zeolite, their radii have to be interpolated according to simple rules.33 Since the halogen ions are located far from the solvated electron, errors made in choosing the width of the halogen charge distribution do not affect substantially the properties of the electron. The width parameter A0 for the oxygen charge is adjusted to give the correct spectrum for the color center in the bromosodalite. This leads to A0 = 0.57 A. We have explored the dependence of the absorption spectrum on various parameters in the electron-frame potential. We found that broader charge distributions for A1 and Si shift the spectrum to the blue without a perceptible change in the intensity, while an increase in the width of the 0 charge distribution has the opposite effect. These shifts are small. We have also found that the spectrum is fairly insensitive to changes of 5% or so in the charges (subject to electroneutrality). Since the spectrum is sensitive to the Naf -electron pseudopotential, we have calculated the spectrum of the Na atom and found the best agreement with the experiments when the crystal radius of Na+ with a coordination number of 6 is chosen for h a .
We note that the pseudopotential equation (2.2) is qualitatively similar to the more refined pseudopotentials described by Bachelet et who use a sum of two such terms in their description of the long-range Coulombic potential for electroncore interactions, and to that used by Sprik and Klein.25 The pseudopotentials used here are nodeless and therefore neglect the so-called “orthogonality hole”.34 It is known from atomic calculations that this type of pseudopotential can place too much of the total charge in the core region and is not norm-conserving. However, as we have repeatedly emphasized, the problems considered here are sensitive mostly to the long-range part of the electron-frame potential, and considering the overall degree of uncertainty in choosing the parameters, it does not seem worthwhile to try using more sophisticated prescriptions for the pseudopotential at this time. Furthermore, the use of normconserving potential^,^^^^^ where the potential within the cutoff radius is essentially a discontinuous step function, causes severe numerical problems. 3. Interaction of Na+ Ions with Framework and Halogen Ions. As we have already mentioned, the spectrum of the solvated electron is sensitive to the positions of the Na+ ions. To calculate the absorption spectrum, we need to determine these positions by minimizing the total ground state energy. To do this, we assume that a Na+ ion interacts with each frame atom through a truncated Rittner37potential17
This is the sum of a short-range exponential repulsion term, a Coulombic term, a dispersion term, and an induced dipole term. The potential of Ooms et al.I7 did not include the last term. We give the frame atoms the same charge qi as in the frameelectron interaction (Table 1). The other parameters are given in Table 2. The effect of the long-range Coulombic interaction is calculated by using the Ewald summation. The Born-Mayer exponential repulsion is used only for the Na’ interaction with oxygen. The A parameter for the Na+-0 interaction is taken from Ooms et al.” while the parameter e in eq 2.3 (needed only for the Na+-O interaction) was adjusted so that the energy minimization for the dry sodalite gives the correct Na+ positions, as determined from X-ray diffraction.8 The optimal value e = 0.284 in the oxygen-Na+ potential was found by using a simplex routine to minimize the force on each Na+ ion. Inclusion of a dispersive and of an induced-dipole term does not improve the quality of the fit. Therefore, we took C and D in the Na+-O interaction to be zero. The value used previously in the literature” for e is 0.305 A. We did not use a short-range repulsive term for the interaction of Na+ with the other frame atoms because the Na+ ions are fairly localized and do not move close to the A1 and Si atoms.
III. Theory and Computational Details We test the proposed potential by calculating the absorption frequency for the solvated electrons in the four materials discussed earlier in the paper. For this we need to know the ground state of the electron and the equilibrium positions of the Na+ ions. In this section we explain how these calculations were performed and point out some procedures that cut down the needed computer power. Since testing potentials involves a very large number of calculations, efficiency-enhancing “tricks” are very important and need to be mentioned even though they do not rely on new computational procedures. 1. Ground State Wave Function. In this calculation we take into account the interaction of the electron with 343 unit cells (about 14 000 atoms), through the potentials described in the previous section. The adiabatic ground state is calculated with a third-order Lanczos method.38 This requires the computation of WIY), n = 1, 2, 3, ..., where H i s the Hamiltonian and IY) can be any normalizable function with a nonzero projection on the ground state. The difficult part is calculating the effect of the kinetic energy operator on IY), and we do this by using a fast Fourier transform (FFT)method. The Lanczos iterations are considered converged when the change in the energy between successive iterations is less than 1 part in lo6 (typically 100 iterations). For the problem considered here this method is 10 times faster than the imaginary time operator method39used by Haug et aL21 Additional computer savings are obtained by using a Gaussian charge distribution (eq 2.1) to describe the interaction of the electron with the frame atoms and the counterions, instead of explicit summation of pair potentials of the Shaw type. The cusp in the Shaw potential causes severe problems with the convergence of its Fourier series: the Fourier coefficients ak of a function that has a cusp converges like k-2.40 Because of this, a faithful Fourier representation of the function requires high wave vectors k. This means a smaller grid spacing and thus a larger number of grid points. When the spectrum is calculated, by a time domain method, high k also means a smaller time step.
Blake et al.
2130 J. Phys. Chem., Vol. 99, No. 7, 1995 TABLE 2: Parameters of the Sodium-Framework Interactionu 0 A1
1224.54
Si
c1 Br I (1
-0.7
0.284
+0.8 155 1.74 1594.62 1595.50
$1.0 -1.0 -1.0 -1.0
0.334 0.347 0.374
The sodium ion charge is le.
The use of Gaussian charge distributions generates further computer time savings if the potential V(r;R) at a space point r, due to the charges located on the frame at R,is obtained by solving the Poisson equation:
V2V(r;R) = - 4 q ( r ; R )
(3.1)
Here Q(r;R)is the total charge density at the point r for a given set of Gaussian charge densities centered around R. Since the total charge density is a sum of these Gaussians, this equation is solved efficiently by using an FIT method, which requires approximately 2MN ln(N) operations, where M is the number of atoms in a cage (approximately 44) and N is the number of grid points (4g3). Using an explicit summation method would take around P M N operations, where 3 represents the number of unit cells considered. The comparative efficiency of the two methods is characterized by the ratio P I 2 ln(N) x 343123 = 14.9 ( J = 7). This indicates that the FFT solution of the Poisson equation provides the potential 15 times faster than a method using the Shaw pseudopotential and an explicit summation algorithm. 2. Equilibrium Position of Na+ Ions Interacting with the Electron. Having generated the electronic ground state, we need to find the equilibrium position of the Na+ ions. One would hope that since the electron is fairly localized in one cage, we might need to optimize only the positions of the ions in that cage and place the others in the positions they have in the dry sodalite. Unfortunately, it tums out that the spectrum is also affected by the positions of the ions in the neighboring cages, and we have to allow the relaxation of 163 ions for the dry sodalite and 216 for the halosodalite. The nuclear displacements leading to the minimum energy are made by solving the classical equations of motion for the Na+ ions, in the force field generated by the frame, the electron, and all the Na+ ions. Newton’s equations are solved with a fourth-order Runge-Kutta algorithm, and the velocities are set to zero every 20 fs. This moves the ions downhill on the potential energy surface. The ground state of the electron is recalculated afte 2-4 nuclear displacements. The minimization is considered successful when the change in the energy between successive steps is less than 1 part in lo7. We check whether the final configuration is a minimum by calculating the force constant matrix of the nuclei and testing whether it is positive definite. We also test whether we have obtained a global energy minimum by using a simulated Monte Carlo annealing to look for other minima. 3. Calculation of the Absorption Spectrum. The absorption spectrum of the solvated electron is calculated with the time-dependent perturbation theory proposed by Heller$l with a m ~ d i f i c a t i o nneeded ~~ when the electron motion is treated explicitly. The details regarding the application of this method to zeolites can be found in Haug et aLZ1 The calculation is performed by allowing the electron to move in three dimensions but keeping the nuclei fixed and by calculating an overlap integral (see ref 21) whose Fourier transform over time gives the spectrum. Keeping the nuclei fixed suppresses the vibra-
tional structure in the spectrum. We introduce into the timedependent formula for the cross section a temporal Gaussian to@^,^^ which gives the absorption line shape a width. This width is artificial, and only the peak frequencies in the calculated spectrum can be compared to the experiment. (The shifts due to nuclear motion turn out to be small.) The computation of the vibrational structure in the spectrum of the solvated electron is a difficult problem because the main candidate for performing such calculations-a combination of a quantum treatment of the electron with a classical treatment of the nuclear motion-tums out to be i n a d e q ~ a t e . ~This ~ problem has recently been s 0 l v e d , 4 ~and * ~ applications to doped sodalites are under way.z2 The time propagation needed for the absorption cross section calculation uses the split time operator method of Fleck et ~ 1 The time step in the calculation is 0.5 au (0.012 fs). The spatial grid extends over 3 unit cells in each of the 4-fold symmetry directions, and 48 grid points are used in each dimension. We propagate the system for 20 fs, and the energy is conserved to within 0.0016%.
IV. Results 1. Background Information. In this section we test the electron-frame and ion-frame potentials by calculating the structure of the dry and halosodalites and the absorption spectrum of an electron solvated in them. First, we summarize the background experimental information regarding these systems. In 1966, Rabo et U Z . ~showed that the dehydrated NaY zeolite tums blue when it is brought in contact with Na vapor. The “dry ~odalite”~ (Na+)3([AlSiO4]-)3, the anion sodalite^'^,^^ (Naf)4( [AlSi04]-)3X- (where X represents a halogen), and the Na+, K+, Rb+, and Csf exchanged zeolites also change color when exposed to alkali vapor, an X-ray beam, or an electron beam. This phenomenon is similar to the formation of color centers in alkali halides49and to the change in color when electrons are solvated in liquid ammonia?O The electron spin resonancelo and the electron spin echolo spectra of these compounds have 13 equidistant hyperfine lines, a result expected for a species involving four equivalent spin 312nuclei. This implies that the color center is the cluster (Na+)4(AlSi04-)3e. The formation of the (Na+)4(AlSi04-)3e clusters in the halosodalites was explained by Stein et U Z . , ~ ~who showed that the halogen ion was missing in about of the cages. It is reasonable to assume that the blue color appears because the electrons produced by the X-ray beam migrated to the missing halogen sites and formed (Na+)d(AlSiO4-)3e. While the color center in all these compounds is the same (Le., (Na+)4(AlSiO4-)3e), Bolwijn et aZ.13have shown that the peak wavelength in the absorption spectrum is a linear function of the host’s lattice constant. By changing the halogen, it is thus possible to change the color of the doped halogen sodalite. The color also depends on the cation. By varying the metal in the metal-exchanged NaY zeolite, Breuer et ~ 1 produced . ~ ~ compounds that spanned the visible spectrum from deep red (in the case of Nay) to deep blue (KY). Finally, the color can also be controlled by varying the time of exposure to the Na vapor.14 The electron-frame and ion-frame potentials developed here must be able to reproduce these observations. 2. Positions of Na+ Ions in Doped Sodalites. In all the calculations performed here we have assumed that the positions of the frame atoms in the doped (Le,,exposed to alkali) sodalites is the same as in the undoped ones. Since we examine here the low dopant concentration limit, this is a reasonable assump-
.
~
~
Electron-Zeolite and Na+-Zeolite Interactions
J. Phys. Chem., Vol. 99, No. 7, 1995 2131
TABLE 3: Cartesian Coordinates for the Seven Na+ Ions Closest to the Electrona atom label
x (bohrs) Y (bohrs) (a) Formal Charge Results -3.94 4.01 -3.94 4.01 -4.48 -4.48 4.49
z (bohrs)
3.98 -3.98 -3.92 4.02 4.62 -4.44 4.53
-3.97 -4.02 3.92 3.97 4.44 -4.65 -4.52
(b) Present Potential -3.69 3.74 3.44 -3.41 3.69 3.73 3.53 -3.47 4.23 4.16 -4.35 -4.42 4.69 -4.40
-3.69 3.42 3.67 3.54 -4.22 -4.70 4.32
TABLE 4: Cartesian Coordinates of the Eight Na+ Ions Closest to the Electron, for the Color Center in the Chlorosodalite" atom label
x (bolus)
Y (bobs)
z (bolus)
1 2 3 4 5 6 7 8
-2.84 2.84 -2.84 2.84 -5.42 -5.42 5.42 5.42
2.84 -2.84 -2.84 2.84 5.42 -5.42 5.42 -5.42
-2.84 -2.84 2.84 2.84 5.42 -5.42 -5.42 5.42
The coordinate system is shown in Figure 1.
TABLE 5: Cartesian Coordinates for the Eight Na+ Ions Closest to the Electron, for the Color Center in the Bromosodalitd atom label
(bohrs) -2.98 2.98 -2.98 2.98 -5.33 -5.33 5.33 5.33
a The origin is the cage center, and the orientation of the axes is as in Figure 1. The coordinates of the frame atoms were taken from ref
8.
tion. The small distortion of the cage containing the electron is likely to have a second-order effect on the spectrum. The spectrum is rather sensitive*l to the positions of the sodium ions interacting most strongly with the electron. If the concentration of the solvated electron is low, these positions cannot be determined experimentally. In the previous work21 the ions were fixed at the positions measured for (Na+)d(OH)-([AlSiO4]-)3, in the hope that the effect of the OH- group on the ions is similar to that of the electron. Here we will determine the Na+ ion positions by minimizing the total energy of the doped zeolite. The positions of the Na+ ions in the cage where the electron is located are most affected by two factors: the oxygen atoms attract the Na+ ions toward the hexagonal windows of the cage; the electron however tends to "bind' the ions, pulling them toward the middle of the cage. Since the ion-ion and electron-ion interactions are long-ranged and since the electron wave function is not completely localized in one cage,*I the presence of the electron will also affect the positions of the ions in the neighboring cages. Because of this we minimize the Na+ ion positions in 26 unit cells (the unit cell has two cages) closest to cage containing the electron; this means that we adjust the positions of 163 Na+ ions for the doped dry sodalite and 216 ions for the halogen sodalites. The procedure used for minimization was described in the previous section. In Table 3 we give the Cartesian coordinates of the seven Na+ ions closest to the electron. These are compared to the results obtained by using the formal charge model. In both models the electron density is largest inside the tetrahedron. This has a bonding effect on the ions forming the tetrahedron and affects their tetrahedral radius (the radial distance from the center of the tetrahedron to any of the vertices). In the formalcharge model the Na+ ion positions hardly relax when the electron is added to the system; in this scheme the oxygen charge is very large, and the sodium ions are pinned near the hexagonal windows where the oxygen atoms are located; the pull of the electron is much too small to affect their positions. In our model the oxygen charges are smaller (Table 1) and as a result the electron can cause the Na+ tetrahedron to shrink. For the doped dry sodalite the tetrahedron radius obtained with the present potential is smaller by 0.32 A than that given by the formalcharge model. We shall see shortly that this difference has a substantial effect on the absorption spectrum. In the case of the halosodalites the electron pulls the Na' ions toward the center of the cage. The tetrahedral radius depends on the
x
a
Y (bohrs)
z (bolus)
2.98 -2.98 -2.98 2.98 5.33 -5.33 5.33 -5.33
-2.98 -2.98 2.98 2.98 5.33 -5.33 -5.33 5.33
The coordinate system is shown in Figure 1.
TABLE 6: Cartesian Coordinates for the Eight Na+ Ions Closest to the Electron, for the Color Center in the IodosodaliW atom label x (bohrs) Y (bobs) z (bolus) 1 2 3 4 5 6 7 8 a
-3.13 3.13 -3.13 3.13 -5.14 -5.14 5.14 5.14
3.13 -3.13 -3.13 3.13 5.14 -5.14 5.14 -5.14
-3.13 -3.13 3.13 3.13 5.14 -5.14 -5.14 5.14
The coordinate system is shown in Figure 1.
TABLE 7: Radial Displacement of the Na+ Ions from Cage Center@ shell 1 (A) shell 2 (A) dry sodalite 3.29 (3.71) 4.02 C1 sodalite 2.60 (2.73) 4.97 Br sodalite 2.73 (2.88) 4.89 I sodalite 2.86 (3.12) 4.71 The values given in parentheses are those for the crystalline host. halogen and is 5.0 bohrs for C1-, to 5.35 bohrs for Br-, and 5.76 bohrs for I- (see Tables 4-7). In all but the C1 case the potential3' gives the same tetrahedron radius (to within 0.01 A) as the experiment. Furthermore, we find that the distance of the second shell of Na' ions from the center of the electroncontaining cage is within 0.04 A of that inferred by Smeulders et ~ 1 . for ' ~ the anion sodalite systems (see Table 7). We expect that the hyperfine coupling constant in these systems will increase from C1 to Br to I, because the nearest-neighbor Na+ ion distance decreases as one goes from C1 to I. This trend is observed in the experiments.1° For the dry sodalite, displacement of the Na+ ions in the cage occupied by the electron is not perfectly tetrahedral, because the electrostatic field generated by the Na+ ions is not perfectly cubic. The mean electron position (Le., the expectation value of the electron coordinates) is in the center of the Na+ tetrahedron, not in the center of the cage.
Blake et al.
2132 J. Phys. Chem., Vol. 99, No. 7, 1995 3.5,
1
I
37
-
Na-doped
2.5
.
3 3 v
2
b
1.5
h
1
0.5
frequency (eV) Figure 2. Calculated absorption cross section for the solvated electron using the formal charge potential for the interaction of the Na+ ion with the zeolite frame. The line is broadened by using a Gaussian window function with a width of 20 fs. Only the peak frequencies are relevant to experiments.
0
1
1.5
2
2.5 3 frequency (eV)
3.5
4
Figure 4. Calculated absorption cross section for the halosodalites. The widths are an artifact of the 20 fs propagation time. 750, 700:
EE wavelength (calculated) SI data 1 Ge data least squares fit
-
650: 600: 550: 500: 1.4 1 . 6 1.8
2
2 . 2 2 . 4 2.6 2.8
frequency (eV)
Figure 3. Calculated absorption cross section for the solvated electron using the potentials proposed in this paper. The measured spectrum (taken from ref 14) is shown for comparison. The width of the theoretical curve is determined by the propagation time for the simulation, which is 7.5 fs. 3. Absorption Spectra. A more stringent test of these potentials is their ability to reproduce the absorption spectra of the solvated electron. First we examine the Na-doped sodalite. The spectrum calculated by using a formal charge Naframework potential is shown in Figure 2. The one obtained by using the potential proposed here is presented in Figure 3, together with the experimental result. The spectrum generated by using formal charges has five peaks, at 0.5, 0.9, 1.3, 1.7, and 2.1 eV. Our potential gives a one-peak spectrum centered around 2.0 eV, in agreement with experiment. The width in the calculated spectrum is determined by the propagation time, so only the peak frequency should be compared to the experiment. In a future article22we will calculate the line width caused by the motion of the Na+ ions. The difference between the two spectra is interesting. We have calculated the excited states of the electron with the formal charge potential and found that the low-frequency peaks (w < 1.5 eV) are due to excited states in which the electron is centered in the cell neighboring the one in which the ground state electron is located (Le., they are charge transfer states). The same states are also present in the calculation using our potential; however, they are absent from the spectrum because they have very little overlap with the ground state. Other details of the spectrum will be given in a later paper.22 The main point is that the results obtained by using formal charges are at variance with the experimental findings, while our results agree with them.
0.7
0.0
0.9
9
9.1
9.2
9.3
(A) Figure 5. A plot of the wavelength of maximum absorption for solvated electrons in sodalitic systems. The squares show the results calculated by us; the other points are experimental (ref 13). lattice constant
TABLE 8: Comparison of the Calculated and Observed Frequencies (in eV) Corresponding to the Maximum Absorption Coefficient' X hwmm(experiment) hwmu(calculated) c1 2.48 2.43 Br 2.25 2.25 I 2.06 2.05 Na-doped dry sodalite 2.00 1.94 a The experimental data for the halosodalites is taken from ref 13 and for the dry sodalite from ref 14.
The results for the anion sodalites are shown in Figure 4 where we plot their frequency-dependent absorption cross sections. The peak wavelength for the halosodalites depends linearly on the lattice constant of the zeolite (Figure 5) as observed experimentally.l 3 Moreover, within experimental error, the predicted peak frequencies are equal the measured ones (see Table 8). The calculated spectra of the electron solvated in halosodalites have a shoulder whose frequency is 1 eV higher than that of the main peak; this too is in agreement with the data.
V. Summary We have proposed a potential for the interaction of an electron and an alkali counterion with sodalite framework. This has been tested by calculating the peak absorption frequency of an
Electron-Zeolite and Na+-Zeolite Interactions electron solvated in the dry sodalite and the chloro-, bromo-, and iodosodalites. The ground state of the electron in these systems is localized around the center of the tetrahedron made by the four sodium ions. The electron is held there by the attraction to the sodium ions and by the repulsion due to the oxygen atoms located on the hexagonal windows. The excited state responsible for the absorption spectrum also happens to be localized. Thus, the absorption spectrum of the electron depends mainly on the interaction of the electron with alkali ions and the oxygen and much less on the charge density associated with the A1 and the Si atoms. The spectrum also depends sensitively on the size of the sodium tetrahedron. This is determined by the location of the oxygen atoms on the hexagonal windows, which in tum depends on the lattice constant. Thus, to first order, the Naf-oxygen interaction has the highest effect on the spectrum. The interaction with the Si and the A1 atoms plays a secondary role. Therefore, there are grounds to believe that the electron-oxygen interaction and the Na+-oxygen interaction are fairly reliably determined in the present work. The potentials used to calculate the spectra had only one parameterthe size of the oxygen charge-which has been fitted to give correct frequency in the spectrum of the electron in the bromosodalite. The charges were provided by ab initio calculations on small clusters. Use of electronegativity to determine these charges leads to comparable results. The size of the charge distribution, for all atoms except oxygen, is given by the crystal ionic radius of the respective atom. The decay length in the Bom-Mayer potential for the Na+-oxygen interaction was determined by fitting the Na+ ion positions in the dry sodalite. In this way, all parameters but one are determined, by specific rules, prior to the calculation of the absorption spectrum. This parameter set gave absorption frequencies, for the electron in chloro- and iodosodalite, in good agreement with experiment. Thus, the prescription used for picking the parameters is supported by the agreement between the calculated and the measured spectra. This is encouraging since the prescription is transferable to other systems, and it is therefore more useful than a set of parameters obtained by fitting data, which would have to be refitted for each new system or data set.
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