SECTION A: CRITIQUE OF THE ONE-CENTRED DESCRIPTION OF THE EXCITATION. It has been suggested1,2 that excitations in alkali halide crystals do not involve charge transfer but that in these processes the halide ion remains in an essentially one-centred but excited state. These one-centred halide ion states were then compared with those of the iso-electronic noble gas so as to obtain a useful description of the coupling between the angular momentum of the np5 halide core with that of the excited electron, especially if this is not considered to be in a halide s orbital. However this model is unable to offer any useful insights in to the energy needed to reach the first excited level which would be described as containing a halide ion in an np5(n+1)s excited state. The fluorides provide the most clear cut example of the failure of this approach. Here the anion excited state is compared directly with that of the 2p53s level of a neon atom. This neon excitation energy of 16.6eV is 6eV larger than experiment for any of the fluorides. Furthermore, if this state is taken to be essentially localized on the fluoride, polarization can hardly be invoked to explain the discrepancy. It is also hard to believe that the magnitude of this polarization energy can be as large as 6eV. This shows that the difference between the nuclear charges of fluorine and neon can not be neglected. However, since the ionization potential of the fluoride ion is 3.4eV, the excitation energy to the 2p53s level of a free F– ion would be even lower. This shows that the one-centred description must introduce the electrostatic potential generated by the surrounding lattice. Since, however, this potential is constant in the spatial region of the fluorine nucleus, introduction of this factor can only affect the predictions if it recognized that the electron, considered as being in the 3s orbital of the fluoride ion, has a significant electron density in the vicinity of the cores of the surrounding cations. In these spatial regions, it experiences both the purely atomic like potential generated by each such core, as experienced by a valence s electron of the metal, as well as experiencing the repulsive electrostatic potential (M-1)/Re of the surrounding lattice. This repulsion, taken in conjunction with the attractive potential –M/Re experienced by a halide electron, shows that the change in the Madelung energy must contribute to Ect. However introduction of these factors introduces all the energetic contributions predicted by the charge transfer description whilst simultaneously removing any purely halogen property other than the electron affinity AH. These considerations show that the halogen centred excitation model in its simplest form yields no insight into the excitation energies whilst refinements introduced to rectify this deficiency lead essentially to the charge transfer description in which the term (2M–1)/Re is the largest contributor to Ect. 1 It has also been pointed out3 that the one-centred np5(n+1)s exciton description of the first excited state does not predict the correct intensities of the two lowest energy transitions. Thus the charge transfer model predicts that the lowest energy transition will appear as a doublet because, after the excitation, the halogen can be left in either the 2P1/2 or 2P3/2 states. In contrast, the lowest three levels of the 2p53s configuration of Ne, having J=0, 1 or 2, belong in the first approximation to the 3P term and are separated by about 400cm-1. The second J=1 level is essentially a singlet lying higher in energy by around 1000cm-1. In the absence of spin-orbit coupling, which will mix a small quantity of the 1P1 state into the 3P1 level, there is only one allowed transition. Since spin-orbit coupling mixes a small quantity of the 1P1 state into the 3P1 level, a second, but much weaker transition will also be observed. As the halogen nuclear charge increases, the second peak would be predicted to increase in intensity until, for the 5p56s configuration of Xe, the spin-orbit coupling is significantly larger than the 5p-6s electrostatic repulsion thereby predicting that the two peaks would have comparable intensities. However the absence of any observed increase in intensity of the second peak, which appears with a much greater intensity in the spectra of the alkali fluorides than would be predicted by comparison with the spectrum of neon, must constitute evidence against the one-centred description. In contrast the charge transfer model predicts, in agreement with experiment, that the relative intensities of the two transitions are essentially independent of the nuclear charge although this latter does govern the energy difference. It should also be noted, as reported previously3-5, that the splitting between the two peaks in each of the spectra of the rock-salt structured materials is very close to that in the spectrum of the corresponding free halogen. This is not inconsistent with the charge transfer description after recognizing that any small discrepancies could well be explained by in-crystal modifications of the electronic structures of the halogens. Consideration of the spectra of the salts having the cesium chloride structure yields3 the same conclusion. SECTION B: COMPARISION OF THE POLARIZATION RESPONSES IN FOUR DIFFERENT CHARGE TRANSFER DESCRIPTONS OF THE EXCITATION. Section IIB outlined the principles and presented the conclusions deduced from approximate calculations of the polarization response of the lattice to the electric fields created by four of the possible types of defect created by the electronic excitation. The lattice, which consists of all the ions still carrying the same charges as in the electronic ground state, was divided into those lying at a distance Re from any defect species plus the remainder (for the forbidden state, the anions located a 2 distance 2 Re from the neutral halogen were also treated as close neighbours). The field created by the defect at each of the former ions was calculated exactly by summing the fields generated by each species in the defect. The remainder of the lattice was treated as a continuum with the defect treated as a point multipole. The (=x, y or z) component, Fs, of the electric field at any ion s located at a distance Re from any defect species was calculated by summing over all the contributions created by each defect species, the latter being labelled t. Fs qt Rts Rts3 (B1) t Here qt is the charge of species t relative to that in the unexcited lattice, so that qt is +1 for the halogen created by the excitation.In (B1) Rts is the component of the vector from species t to ion s. The resulting fields at each type of lattice ion, as illustrated in Figure 2, are presented in Table SI for each of the four possible defects. The quantities reported as (1/2)|F(S)|2 in the rows labelled ‘Total’ were calculated as the sum of the products of the individual anion (1/2)|F(S)|2 multiplied by the degeneracies listed in the last column. Each of these latter is the number of ions related by symmetry to that listed in the second column. These total (1/2)|F(S)|2 appear in the first column of Table III where they yield the polarization energy of the near neighbour anions as –A(1/2)|FA2| in the approximation in which the interaction between the induced dipoles is neglected. For the cations in the dipolar and symmetric defects, the mean square fields and degeneracies are not reported in Table SI as symmetry demands that they are the same as the corresponding quantities for the anions. For the quadrupolar and forbidden defects, the coefficient of each cation contribution to the polarization energies reported in Table III was derived as the product of the cation (1/2)|F(S)| with the degeneracy. In the continuum description of the ions located at distances greater than Re from any species pol in the defect, their contribution E crdis to the lattice polarization energy is given in the approximation of neglect of interactions between the induced dipoles by 1 pol E crdis | F(R) |2 dV 2 (B2) Here |F(R)|2 is the square modulus of the electric field created at position R by the defect and is the density of polarizable material given by cr/Vm which reduces to cr/(2 R e3 ) for rock-salt structured crystals. The origin is taken at the centre of the defect so that the volume integration in spherical polar coordinates is over both the angles and with the radial integration extending from some 3 minimum Rm to infinity. Each defect is treated as a point multipole with its components presented in the first two lines of Table SII. The three resulting components F of the electric fields presented in lines 5 to 7 of this table were derived from the expression appearing in the third line by using the relations (41) and (36) of the review6. The polarization energies presented in the ninth line of Table SII were derived from equation (B2) after performing the integrations described above. The lower limit Rm of each radial integration was taken to be the distance from the origin to the closest ion not considered amongst the defect nearest neighbours whose contributions appear in Table SI. In the last pol line of Table SII, the final polarization energies E crdis arising from the ions treated as a continuum were calculated by using the Rm values appearing in the previous line. The numerical contributions to pol pol E crdis appear in decimal form in Table III. For the defect labelled forbidden, the value of E crdis is twice that for the quadrupolar defect because the unique quadrupole moment component of the former is twice that of the latter, both quadrupoles being traceless with only two different non-zero numerical values. SECTION C: POTENTIAL PARAMETERS AND CALCULATIONAL DETAILS 1. Polarizabilities and dispersion energies The dispersive attraction between mononuclear species X and Y was evaluated as a function of their internuclear separation rXY through disp 6 8 QD 8 VXY (rXY ) 6 (dX ,dY ;rXY )C6 (XY)rXY 8DQ (dX ,dY ;rXY )C8DQ (XY)rXY QD 8 (dX ,dY ;rXY )C8 (XY)rXY (C1) Here C6(XY), C8DQ (XY) and C8QD (XY) are respectively dipole-dipole, dipole-quadrupole and quadrupole-dipole dispersion coefficients. The parameters dX and dY derived from the properties of the individual species X and Y define the function n(dX,dY;rXY) which reduces the magnitude of the undamped (n(dX,dY;rXY) =1) attraction at rXY values at which the overlap of the electron densities of the interacting pair is not negligible. The Axilrod-Teller triple dispersion interactions make only very small (c.a. 5kJ mole-1) energy contributions7 and were not included as their consideration would very significantly increase the computational requirements. The dipole-dipole C6(XY) coefficients were calculated from the polarizabilities via the SlaterKirkwood formula8 4 C6 (XY) 3 X Y 1/ 2 2 ( X / PX ) (Y / PY )1/ 2 (C2) For each cation and anion, the electron number PX was derived9 by demanding that this formula reproduced the exactly known C6(XX) for the iso-electronic noble gas from the exact polarizability of that gas. The required noble gas data are assembled in Table SIII. Each alkali atom electron number PM presented in this table, derived using (C2) from accurately known values of C6(MM) and M, differs very slightly from that derived previously9 because the present calculations used newer and very slightly different values for the alkali atom polarizabilities M. The testing9 of (C2) by comparing the best currently available values with its predictions for the coefficients of both unlike alkali atom pairs and alkali atom noble gas pairs was repeated using the present M and PM. This comparison does not need to be presented here in detail because the present Slater-Kirkwood predictions differ from the results appearing in Table A2.29 by at most 1%. The dipole-quadrupole dispersion coefficients were calculated by scaling the predictions of Starkschall-Gordon formula10 3 C8DQ (XY) C6 (XY) r 4 2 Y / r2 Y (C3) Here r2Y and r4Y are the electronic expectation values calculated by considering just the most Y as shown to yield the most reliable predictions9,11. Thus only the six loosely bound electrons of ion p electrons in species of s2p6 outermost electronic configuration are included whilst for an alkali atom only the valence s electron is considered. However, for the interactions of two species having outermost p6 electronic configurations, even this direct use of (C3) predicts values that are between 20% and 30% too small11. Each prediction of (C3) was therefore multiplied by a scaling factor derived from a closely related pair for which a more accurate result12,13 is available. For interactions involving solely the ions, each scaling factor K(CC), K(AA), KDQ(CA) and KQD(CA), in an obvious notation, of 1.403, 1.285, 1.352 and 1.394 was derived from NaCl as the best ab initio quantum chemistry result divided by the corresponding prediction from (C3). Current evidence indicates that these scaling factors for species of p6 outermost electronic configuration are insensitive to the nuclear charges11-14. For the interactions between an alkali metal and a noble gas, the accuracy of the Starkschall-Gordon predictions (C3) was tested (Table SIV) by the comparing these with the currently most reliable values. The results show that these predictions are as accurate as those for those of the interactions between two species of p6 outermost electronic configurations. This justifies 5 using for each metal-ion interaction the scaling factor (Table SV) derived as the ratio of the best to the Starkschall-Gordon prediction for the interaction of that metal with the noble gas iso-electronic with the ion. The mathematical forms9,11,15 of the dispersion damping functions in (C1) have been conveniently summarized7. The damping parameters (dX) derived using the previously presented methods11, have already been presented for all the anions7,16, excepting those in KCl, KBr and RbBr, all the cations7,16,17 and the nanotube carbon atoms18. For the three latter crystals, the parameters will be reported elsewhere. For the neutral metal atoms the damping parameters (dM, Table SVI) were calculated from the result of Lassetre theory19 also used for the cations and carbon atom dM 2 I M IM IM (C4) Here all quantities are expressed in atomic units with IM being the energy of the longest wavelength dipole allowed transition. The required energies were taken from Moore20. All the dispersion coefficients involving purely inter-ionic7,16,17 or ion-carbon18 interactions have already been presented excepting those involving the anions in KCl, KBr and RbBr which will be reported elsewhere. The coefficients for the interaction of each metal with both its cation and a carbon atom are presented in Table SVI whilst Table SVII reports those for the interaction of each metal with its in-crystal anion. In the computations using the GULP program, the shell model used to describe the polarization responses requires for each species X, a shell charge (YX) and a quadratic force constant kX, which determine the polarizability X as YX2 / k X . For the cations, metal and nanotube carbon atoms, the natural choice for YX (–PX) of the negative of the electron number for each shell charge allows kA to determined by demanding that the polarizability is reproduced (see Table I, Table SIII, and footnote b to Table SVI). For the in-crystal anions, the parameters YA and kA were determined, as described elsewhere16 for RbCl, by demanding that the experimental values of both the high frequency and low frequency dielectric constants of the bulk crystals are simultaneously reproduced. The details for both the neutral halogens and the other crystal ions will be reported elsewhere. 2. Short range interactions The short range inter-ionic potentials needed for the GULP computations were derived using the RELCRION program as described elsewhere16. Since the resulting potentials do not describe any effects arising from electron correlation, each such potential was augmented9,11,16 by an electron 6 correlation contribution of short range computed from the electron densities of the ions using the density functional theory of an electron gas of uniform density21. The model, the OHSMFS model11, used to describe the significant modifications of the anion wavefunctions by their environment incrystal causes the total crystal energy to contain, in addition to the usual two-body interactions, the rearrangement energy9,22 Ere(R) needed to convert a free anion into its form optimal for the crystal having geometry defined the closest cation-anion distance R. Since, however, the GULP program does not handle compressible ion models and rearrangement energies, it was necessary to introduce eff (R) ) defined by for each of the cubic phases, the short range cation-anion effective pair potential ( VsCA eff VsCA (R) VsCA (R) 1 E re (R) nCA (C5) where nCA is the coordination number. This definition ensures17,23 that the expression (2.3) of reference 21 for the total crystal cohesion UL(R) is reproduced if (C5) is taken to be the short range cation-anion interaction in standard expressions for the crystal energy which do not contain a rearrangement term. The GULP program can handle two-body potentials in both numerical and a range of analytic forms. Since these potentials have already been computed for both RbCl16 and all three of the iodides7 considered here, the only interactions that needed to be computed were those involving the anions in KCl, KBr and RbBr. For each of these three crystals, the computed potentials and their representation by simple analytic functions will be reported elsewhere. For the encapsulated crystals of KI, the potentials describing the interaction between each ion and the carbon atoms of the nanotube wall had to be computed for the previous study18 of ground state encapsulated crystal geometries. The short range interactions were computed from the HartreeFock electron densities21,24 using the density functional theory of an electron gas of uniform density and finite extent25,26 in the variant27 as fully discussed elsewhere22. 3. Computational techniques Since the ions in an encapsulated crystal do not lie on centres of symmetry, these species will be polarized in even the ground state of the entire system of tube plus crystal. If the computation is performed for a geometry, which does not minimize the total energy in the ground state, there will be non-zero forces acting on the shells which will therefore acquire additional displacements and energy contributions which are spurious in that they do not arise at the equilibrium geometry predicted by the computations. 7 The above difficulty was handled by first calculating the energy of the ground state at the optimal nuclear positions with all ions carrying their charges of unit magnitude and neutral carbon atoms. For the case of square planes, these nuclear positions were generated from the experimental values of a and b whilst, for the diamond shaped planes predicted by the computations, each nuclear geometry was taken from the latter. The omission of any shell model shells ensured that all the species were unpolarizable. This gives the electrostatic, short range and dispersive contributions to the excitation energy. Then the shells are added to the ions still retaining their charges of unit magnitude with still neutral carbons and the shells in the ground state relaxed to equilibrium at the same nuclear geometry. The excitation energy to reach the excited state is then computed with the shells retaining their displacements previously found to be optimal for the ground state. This calculation is then repeated after relaxing the shells to their positions optimal for the excited state. The difference in the excitation energies predicted by the two latter calculations is the required polarization energy. This polarization energy, considered as the best approximation, is added to the electrostatic, short range and dispersion contributions predicted at the experimental geometry to yield the final prediction for the excitation energy. A computation of the excitation energy, at a nuclear geometry not yielding the minimum energy predicted from the potential model used, will predict an excited state polarization energy containing a spurious contribution from forces arising from the inconsistency between the potential model and the fixed nuclear geometry. The fully ionic computation for the ground states of some of the crystals, including all of the iodides, predicted7 internuclear separations that were greater than experiment by around 0.15a.u. The need to perform the excitation energy computations using potentials predicting the experimental nuclear geometry to a much greater precision necessitated introducing the small additional attractive cation-anion interaction defined by equation (3.1) of reference 7. For each crystal, the two parameters defining this potential were derived by demanding that both the experimental equilibrium separation and bulk compressibility were reproduced. The need to introduce this small additional potential is shown by the results, derived without its consideration, of 8.40eV, 7.97eV, 7.52eV and 7.35eV for the excitation energies of KCl, RbCl, KBr and RbBr respectively computed, for the reasons presented above, at the equilibrium geometries predicted using the fully ionic model. These results are significantly greater than both experiment and the predictions presented in Table IV. These energies are greater than experiment because the positive purely coulombic ([2M–1]/Re) electrostatic contribution decreases with increasing internuclear distance more rapidly than the magnitudes of the negative dispersion and polarization contributions. However it should be noted that the small additional potential, being taken to be the 8 same in the ground and excited states, does not very directly affect the predicted excitation energy. The only role played by this potential in the determination of Ect is that of making a minor contribution to the changes in the inter-shell interaction energies arising when the shells in the excited state are displaced from their ground state equilibrium positions. 9 TABLE SI. Electric fields created at nearest neighbour ions by four types of defect a-c Defect dipolar ion A1` A2 total quadrupolar Fx(S ) –3/4 1/(2 2 ) Fy(S ) Fz(S ) (1/2)|F(S)|2 0 0 0.2813 1 0 0.2715 4 –[1–1/(2 2 )] 1.367 A1 –0.1172 A2 0 0 0.0069 4 0.0365 0.0365 0 0.0013 4 A3 0.2126 0 0.0227 8 C1 0 0 0.2089 2 –0.0150 1–1/(2 2 ) total symmeterical 0.2144 A1 –0.0544 A2 –0.0780 0 –0.0780 0 0.0015 6 0 0.0061 12 total forbidden deg A1 0.0822 –11/36 0 –0.1912 0 0.0467 2 0 0.0531 8 A2 0.2641 A3 0 0.1611 0.1611 0.0260 4 C1 0 1–1/(2 2 ) 0 0.2089 4 total 0.6224 a absolute values of fields F(=x,y or z) given by multiplying the scaled F (S ) values by 1/Re2 b absolute values of mean square fields |F|2 given by multiplying the |F(S)|2 results by 1/Re4 c the quantities labelled “total” are defined in the text. 10 TABLE SII. Continuum descriptions of polarization energies of distant ions. dipolar quadrupolar y = z =0 = 0 (≠) x = – R e/2 –2xx =–2xx = zz = R e2 /2 F(D ) = ∑ T F(Q ) = ∑ ∑T T = 3R-5( – R2) -7 T = –3R [5 –R2)(++)] Fx(D ) = xR-5(3x2–R2) Fx(Q ) =(3/2)zzR-7x(5z2–R2) Fy(D ) =3 xR-5xy Fy(Q ) =(3/2)zzR-7y(5z2–R2) Fz(D)=3 xR-5xz Fz(Q) =(3/2)zzR-7z(5z2–3R2) |F(D)|2 = x2 R-6(3sin2cos2 +1) |F(Q)|2 =(9/4) 2zz R-8(5cos4–2cos2 +1) pol E crdis = –(1/2)cr4/(3Re Rm3 ) pol E crdis = –(1/2)cr3 /(10Re Rm5 ) Rm=( 13 /2)Re Rm= 3 Re pol E crdis = –cr Re4[16 /(39 13 )] pol E crdis = –cr Re4 [ /(60 3 )] TABLE SIII. Polarizabilities, C6(XX) coefficients and electron numbers for alkali atoms and noble gases (a.u.) a,b,c,d X Li Na K Rb Cs Ne Ar Kr Xe X 164.0 162.7 292.9 319.2 402.2 2.676 11.096 16.753 27.318 C6(XX) 1390 1580 3820 4600 7380 6.93 PX 0.7787 1.030 1.032 1.157 1.488 4.455 68.5 6.106 139 301 7.305 7.901 a X = alkali metal M or noble gas N. b Na polarizability from atomic interferometary experiment28, other alkali polarizabilities from E-H balance experiments29 c see footnotes to Table A2.1 of reference 9 for the primary sources of noble gas data d alkali atom C6(XX) from reference 30. 11 TABLE SIV. Comparison of best and Starkschall-Gordon predictions for dipole-quadrupole dispersion coeffcients (a.u.) a,b C8DQ (NM) C8DQ (NM) NM USG TNC USG NM USG NeLi NeNa 2062.14 205.781 263 2647.06 1910 2380 233.473 285 ArLi 8234.21 ArNa 10534.09 NeK 5141.27 2380 233.473 285 ArK NeRb 6258.57 6460 351.881 NeCs 9291.26 9360 KrLi C8QD (NM) TNC USG 7360 9200 1630.73 1910 1843.93 2061 20649.87 21000 2538.49 3210 554 ArRb 25119.19 25300 2802.85 3880 447.85 675 ArCs 37274.02 36800 3565.63 4700 12301.08 10800 3040.85 3070 XeLi 19640.58 16500 6342.84 4930 KrNa 15714.52 13500 3433.52 3320 XeNa 25014.40 20600 7140.14 5330 KrK 30901.57 3100 4741.64 5150 XeK 49454.85 47800 9913.68 8270 KrRb 37585.37 37400 5234.85 6170 XeRb 60139.99 57600 10942.75 9900 KrCs 55757.50 54500 6657.70 7470 XeCs 89208.03 84400 13915.62 12000 a TNC C8QD (NM) TNC USG, predictions of Starkschall-Gordon formula (14) derived using the Slater-Kirkwood results for the C6(NM) coefficients and radial expectation values taken from reference 31. b TNC values taken from Table IV of reference 32 derived from Pade approximant analyses of experimental oscillator strength data. TABLE SV. Starkschall-Gordon scaling factors for metal-noble gas (MN) interactions KAr KKr KXe RbAr RbKr RbXe CsXe KDQ(MN) 1.265 1.086 0.834 1.384 1.179 0.905 0.862 KQD(MN) 1.017 1.003 0.967 1.007 0.995 0.958 0.946 12 TABLE SVI. Metal dependent dispersion coefficients and damping parameters (a.u.)a-d dM C6(MC) C8DQ (MC) C8QD (MC) C6(MW) C8DQ (MW) C8QD (MW) K 1.0125 131.194 1075.8 10340.9 226.79 3073.3 17481.2 Rb 0.9910 0.9604 2449.9 5143.8 20658.5 Cs 244.494 516.979 317.92 4308.2 31491.4 a 48443.6 Cn(MW) is the Cn(XY) involving the interaction of a neutral metal atom (M) with a nanotube carbon (W). b C6(MX) calculated from the Slater-Kirkwod formula (13) from the polarizabilities in Tables I and II and electron numbers in Table SIII using18 W=10.26a.u. and PW = 1.118. c C8DQ (MX) and C8QD (MX) calculated from Starkschall-Gordon formula (14) using the scaling factors presented in Table SV and metal valence orbital radial expectation values from reference 30. 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