J. Phys. Chem. 1992, 96,9781-9787 (23) Pacios, L. F.; Christiansen, P. A. J. Chem. Phys. 1985, 82, 2664. (24) Hurley, M. M.; Pacics, L. F.; Christiansen, P. A.; Ross, R. B.; Ermler, W. C. J . Chem. Phys. 19%6,84, 6840. (25) LaJohn, L. A.; Christiansen, P. A.; Ross, R. B.; Atashroo, T.; Ermler, W. C. J. Chem. Phys. 1987,87, 2812. (26) Dunning, T. H.; Hay, P. J. In Modern Theoretical Chemistry; Schaefer, H. F., Ed.; Plenum: New York, 1976; p 1. (27) Huzinaga, S . J . Chem. Phys. 1%5,42, 1293. (28) Johns, J. W. C. J . Mol. Spectrosc. 1984, 106, 124. (29) Rogers, S.A.; Brazier, C. R.; Bernath, P. F. J . Chem. Phys. 1987, 87, 159. (30) Wallace, N. M.; Blaudeau, J. P.; Pitzer, R. M. Inr. J . Quantum Chem. 1991.40, 789. (31) GAUSSIAN 90, Revision F Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foreman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M.; Binkley, J. S.;Gonzales, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R.L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.;Pople, J. A. Gaussian Inc., Pittsburgh, PA, 1990.
9781
(32) See Table 7 in ref 18. (33) Botchwina, P.; Sebald, P.; Burmeister, R. J . Chem. Phys. 1988, 88, 5246. (34) Wadt, W. R. J. Chem. Phys. 1978,68, 402. (35) Aziz, R. A. J . Chem. Phys. 1976,65, 490. (36) Bondi, D. K.; Connor, J. N. L.; Manz, J.; RBmelt, J. Molec. Phys. 1983, 50, 467. (37) Dehmer, P. M.; Dehmer, J. L. J . Chem. Phys. 1978, 69, 125. (38) Schwentner, N.; Koch, E.-E.; Jortner, J. Electronic Excitations in Condensed Rare Gases; Springer-Verlag: Berlin, 1985; p 27. (39) Gough, D. W.; Smith, E.B.; Maitland, G. C. Molec. Phys. 1974,27, 867. (40) Kunttu, H.; Feld, J.; Alimi, R.; Becker, A.; Apkarian, V. A. J. Chem. Phys. 1990, 92,4865. (41) van Biesen, J. J. H.; Stovis, F. A.; van Veen, E. H.; van den Meijdenberg, C. J. N. Physica 1980, IOOA, 375. (42) Weaver, A.; Metz, R. B.; Bradforth, S.E.; Neumark, D. M. J . Phys. Chem. 1988. 92, 5558.
Interactions between Cgoand Endohedral Alkali Atoms B. I. Dunlap,* Theoretical Chemistry Section, Code 6179, Naval Research Laboratory, Washington, D.C. 20375-5320
J. L. Ballester, Department of Physics, Emporia State University, Emporia, Kansas 66801
and P. P. Schmidt Chemistry and Materials Division, Office of Naval Research, Arlington, Virginia 22217-5000 (Received: June 22, 1992; In Final Form: September 11. 1992)
All-electron local-density-functional (LDF) total-energy calculations are used to study the interaction between icosahedral C , and endohedral lithium, sodium,and potassium ions and atoms. LDF potential energies as a function of radial displacement , suggest that the carbon shellalkali interaction is spherically symmetric from the center along the 5-fold and 3-fold axes of C to a good approximation. At equilibrium Li’, Na’, and K+ are displaced radially outward 1.4,0.7, and 0.0 A, respectively, from the center of the ball in the ground states of both the neutral and positively-charged complexes. Excited intramolecular chargetransfer states of the neutral molecules exist in which the endohedral alkali ion is neutralized. For these electronically excited neutral molecules the equilibrium position of the alkali atom is at or very near the center of the C, shell. A minimum in the spacing between totally-symmetric endohedral vibrational energy levels indicates a potential energy maximum at the center of the shell. The height of the potential energy maximum lies between the two corresponding vibrational energies.
I. Introduction At the fmt recognition that icosahedral Cso was experimentally important, the molecule was described as a ball and postulated to be able to trap atoms inside.’ Soon experimental evidence was found for trapping a lanthanum atom inside.2 The fact that certain atom-fullerene complexes could loose C2units only down to some endohedral-atom-dependentsize was further evidence that atoms could be trapped inside C, and C70.3With the advent of a method for making macroscopic amounts of these fullerenes: gas-phase collisions of Cso and C70with helium were shown to result in a helium atom being trapped i n ~ i d e . ~Collisions -~ can also trap a neon atom in bu~kminsterfullerene.~~~ Fullerenes with a single lanthanum atom inside (La@Cs2)10 and a dimer inside (La2@C8,,)I1 have been directly synthesized in macroscopic amounts. The electron-spin-resonance(ESR) hyperfine spectra of suggest a formal oxidative state La3+C823-for the molecule.12 X-ray photoelectron and ESR spectroscopies support this assignment and a similar assignment for Y@c82 in work that also suggests that other endohedral Y,C, species including Y&82 may be extractable.” These findings for ytterbium endohedral fullerene complexes have likewise been confirmed.I4 Given that both Y2c82 and L2c82 have been made, one would expect endo-
* Address correspondence to this author. 0022-365419212096-9781$03.00/0
hedral YLaCs2 to be synthesized, which it has.I5 Remarkably, the internal volume of the fullerenes is large enough that endohedral %&2 can also be made.’6J7 Studies of chemical reaction with oxygen show that these complexes are indeed endohedral.I8 Apparently C, with scandium, ytterbium, or lanthanum entrapped is too unstable or created in such low abundance that it cannot be extracted; however, endohedral FeCm can apparently be.2o While all fullerenes can apparently accept endohedral atoms, the Cso endohedral complex is most fascinating because Csois in many respects well approximated by a spherical ball. Aspects of its electronic structure can be viewed as arising from a potential that is slightly perturbed from having spherical ~ymmetry.~J~ The nodal structure of Csomolecular orbitals is strikingly similar to the nodal structure of spherical harmonics.22 It is natural to consider a single endohedral atom placed at the center of this approximate ball. The electronic response and structure of various atoms at the center of Cm have been computed by a number of worker~.~~-~I On the other hand, graphite is known to bind many different kinds of atoms including alkali atoms. Lithium-, sodium-, and potassium-intercalatedgraphite compounds can be made in which the alkali atoms not only chemisorb but migrate between the graphite sheet^.^^^^^ Photoemission studies of the initial stages of intercalation (alkali evaporation onto a 100 K graphite surface) 0 1992 American Chemical Society
9782 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
suggest that potassium forms an ordered array on the surface and donates one electron per alkali atom to graphite, that sodium neither forms an ordered overlayer nor donates charge, and that lithium immediately goes into the bulk.34 All three of these alkali-intercalated graphite compounds s~perconduct.~~ These facts suggest a strong interaction between graphite and alkali atoms. To the extent that Cm approximates a curved sheet of graphite, therefore, one might expect that an endohedral alkali atom would be attracted toward the inside surface of the Cm ball and that a sodium atom would keep its valence electron while potassium would donate its valence electron to the ball. The existence of these intercalation compounds could also suggest facile motion of the alkali atom on the inside surface (and outside surface for that matterz9)of Cb0. Cioslowski and Fleishmann2' found that Na+ occupies a position of unstable equilibrium at the center of Ca. In particular, a 4-3 1G basis set Hartree-Fock (HF) calculation on CsoNa+gave the minimum energy when the sodium ion is displaced 0.66 A from the center. The energy lowering was small, 0.04 eV, and independent of the angular orientation of c 6 0 relative to the displacement vector. Recently, larger basis set (double-r plus polarization) HF calculations give equilibrium displacementsof 0.574 and 1.297 A from the center and energy lowerings of 0.02 and 0.31 eV for endohedral Na' and Lit, respectively.28The offcenter equilibrium position for an endohedral sodium ion was confirmed in linear combination of Gaussian-type orbital (LCGTO) localdensity-functional (LDF) calculations on CaNa+.30 On the basis of MNDO calculations, Bakowies and ThielZ9conclude that Li+ moves rather freely on a 1.4-A sphere concentric with the C a shell, but that this equilibrium endohedral sphere is energetically more than an electron volt uphill compared to a separated lithium ion and thus even more uphill from a chemisorbed lithium ion on the outside of the shell. To the extent that c 6 0 can be viewed as a ball on which the valence electrons are evenly distributed or perfectly free to respond to applied fields, one would not expect its degree of ionization to affect the motion of an endohedral ion, by Gauss's law of electrostatics. Graphite is a semimetal, and Cm was postulated to have an unusually large magnetic response.' Attributed to an unusual cancellation between the diamagnetic and paramagnetic contributions, the magnetic susceptibility was predicted36and m e a ~ u e r dto~be ~ very small. Thus, it is unclear a priori how the charge of the complex should affect internal motion. An approximate theory of the interaction between the endohedral ion and the Cm shell has been formulated on the basis of static (Hartree-Fock) Cm polarizabilitie~.~~ In LDF models additional dynamic (van der Waals) interactions are modeled by effects due to overlapping charge distributions. In this work we use the LCGTO LDF method to map out the potential energy surfaces of endohedral lithium, sodium, and potassium ions along high symmetry directions. The variations of the total energy are small enough that some analysis of numerical procedures was undertaken. In the next section the orbital and two auxiliary GTO basis sets that were used are described. The third section concerns numerical effects on the precision of these surfaces. The fourth section describes these surfaces, and the fifth section discusses fitting these surfaces and describes qualitatively the vibrations of the molecule as a whole. 11. LDFIkrsiSets The LDF used in this work is the Perdew-Zunger (PZ)39 functional form for the exchange-correlation (XC) potential that interpolates between the essentially exact Ceperley-Aldera free-electron gas calculations in the completely ferromagnetic and completely paramagnetic limits. The starting LCGTO basis set ~ ' augmented with a d exponent for carbon, an 1 ls/7p b a ~ i s , was of 0.6 bohr-2.4z The carbon orbital basis set was contracted 2,3/ 1,2/0,1, where for each angular momentum the number of contracted atomic functions and uncontracted diffuse Gaussians are separated by a comma. This same set of primitive orbital exponents and contraction scheme for carbon have been used in a number of LDF studies: The LDF geometries of c60, C60H60,
Dunlap et al. CEoNa+LDF nonvariational
0.4
1
0.0
0.2
0.4
0.0
Cmtar-Sodium M.t.nce
0.8
1.0
(-om)
Figure 1. LCGTO LDF potential-energy curves for the sodium ion in C&a' along any of the 5-fold axes (solid line) or the 3-fold ax= (dashed tine) as computed without treating the XC energy density The energies are relative to the energy when the ion occupies the central pition. The computed relative energies are connected by straight lines.
and CmFm have been 0ptimized.4~The LDF electronic structure and total energies at a number of different geometries have been computed for C44;44Cso,C70r c72, c76, and C84;45and Clg0and Cz.,,.46 The starting LCGTO basis set for lithium, a 11s basis?' was augmented with a p exponent of 0.076 The lithium orbital basis set was contracted 1,6 according to a spin-restricted (same orbitals for different spins) calculation on the neutral atom. The starting LCGTO basis set for sodium, a 12s/6p was augmented with a p exponent of 0.061 bohr-2.42 The sodium orbital basis set was contracted 2,3/1,3 according to a spin-restricted calculation on the neutral atom. The starting LCGTO basis set for potassium, a 14s/lOp was augmented with a p exponent of 0.061 bohr-z.42 The potassium orbital basis set was contracted 3,3/2,3 according to a spin-restricted calculation on the neutral atom. The 11s carbon orbital basis functions, the 11s lithium orbital basis functions, the 12s sodium orbital basis functions, and the 14s potassium orbital basis functions were scaled by 2 and z/3 to generate the atom centered, s-type parts of the charge density and XC fitting basis, respectively.a For sodium and potassium only, all even-numbered porbital exponents (starting from the smallest) were also doubled to generate 9 Gaussian atom-centered functions with which to fit the charge density. No contractions of these auxiliary functions were made. This contrasts with earlier work in which the six largest s-type charge density fitting exponents for sodium were contracted according to the neutral atom calculati~n.~~ To fit angular variations around each atom, 3p and 3d fitting exponents of 0.1,0.3, and 0.9 bohr-z were used on the lithium atom, Sp and 5d fitting exponents of 0.25,0.37,0.7,2.0, and 10.0 were used on the sodium atom, and 5p and 5d fitting exponents of 0.04,0.2,0.6,2.0, and 10.0 b o h F were used on the potassium atom. On each carbon-carbon nearest-neighbor bond an s Gaussian function was used with exponents 1.0 and 0.33 bohr-* in the charge density and XC basis sets, respectively.
III. Minimizing " e r i d Sampling Effects In previous LCGTO LDF work the binding energy of the sodium ion in CmNa+was found to increase 0.12 eV upon being displaced 0.7 A from the center along a S-fold axis and to gain the same amount of energy upon being displaced 0.6 A along a 3-fold axis.M Those two potential energy curves are superimposed in Figure 1. The curves are the LDF energies for various displacements along the two high-symmetry axes relative to the LDF energy when the displacements are zero. In these calculations the 60carbon atoms are fmed at their optimized positions in empty Cm43and the sodium ion is stepped along two axes in increments of 0.1 A or more. Thw relative energies are COM& by straight lines in the figure. An indication of uncertainty in these calculations is the fact that the 3-fold curve is constant to within a ten-thousandth of a Hartree at 0.4,0.5, and 0.6 A. Nonsmooth
Cm and Endohedral Alkali Atom Interactions C,,Na (3-Fold Axis) LDF
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9783
TABLE I: Totd Perdew-Zmger LDF Jhergies (in Hutreea) for u+,Nd, .ad K+ M t e d 8t the center of I c o " l Cu As Computed by Three Different Subsymmetries of the Icoerhcdril Group
symmetry
C&i+ 2427.9541 2427.9583 2421.9620
Ih
6% G"
CmNa+ 2213.8584
CmK+ 2864.1316 2864.1419
2273.8610 2273.8683
C,,Na (5-Fold Axis) LDF
0.0
0.2
0.4
0.6
0.8
0.4
1.0
1
11'
Center-Sodium Distance (h&romi)
Flgwe 2. Variational XC potential-energycurves for sodium in Cmalong
any 3-fold axis. The solid curve is the ground state of the positive ion CmNa+.The similarly shaped curve with long spaces between dashes is the ground state of Cm-Na+. The very different curve, with its equilib rium position at the center, is the intermolecular charge-transfer state CmONao.The true displacement in energy between these curves must also include the quantities given in Table 11.
h
r
x
0.2
$
0.1
2
3
0.0 -0.1
I
potential energy surfaces are a sign of inaccuracy in treating, necessarily by numerical sampling, the PZ XC potential.49 Compared to the two other density functionals in the LDF formalism, the electron-nucleus and the electron-electron Coulombic repulsion, that also are iterated to self-consistency, the XC density functional has the weakest dependence on the density; thus treating is requires fewer sampling points than would be required to sample any other term in the LDF total-energy expression. Nevertheless the sampling scheme is important. About each atom sampling points are generated by using a radial logarithmic grid and an evenly distributed angular grid. The interval between points in the radial grid doubles at every fortieth point and the angular grid is a single point within half of the appropriate Slater atomic radius and 12 points outside that distancesoAt each radius the angular shell is rotated rand~mly.~' The distance from each atom at which the radial grid terminates depends on machine precision and the smallest orbital exponent. For these calculations there are roughly 900 grid points per symmetry-inequivalent atom. Highly efficient recursive, analytic computation of all remaining (nonXC) solid-spherical-harmonic Gaussian integrals for arbitrary angular momenta5*means that the number of sampling points must be minimized. Sampling is done only about symmetry-inequivalent atoms, the approximate volume elements associated with sampling points are reduced rapidly and linearly to zero as points get closer to another atomic center,"9 and the weighting functions are an optimized function of the self-consistent densityeso There are two symmetry-inequivalent atoms in the endohedral atom problem if the atom is at the center. There are nine symmetry-inequivalent atoms if the endohedral atom lies along a 5-fold axis, and there are 13 symmetry-inequivalent atoms if the endohedral atom lies along a 3-fold axis. Except for some deviation with icosahedral symmetry, the cost of all calculations in this work scales linearly with this number of symmetry-inequivalentatoms. Thus numerical sampling of the XC is by far the major component of any LCGTO LDF calculation except on a molecule that has extremely high symmetry. A variational method has recently been described for fitting the XC potential.'3-54 In it each electron moves in a nonlocal potential that is the variation of the fitted XC energy with respect to its molecular orbital at each point, thus the resultant LDF energy is necessarily lower than a calculation in which the electrons move in a fit to the local potential given by the LCGTO density. This improvement together with uncontracting the core of the fit to the charge density of the sodium atom gives the relative potential energy curves along the 3-fold axis depicted in Figure 2. Now the solid curve will almost exactly overlay the solid curve of Figure 1, suggesting a much more spherically symmetry endohedral potential energy surface than is suggested by Figure 1. A better way to analyze the precision of these calculations is to consider the total all-electron LCGTO LDF energy of a central
i
0.0
1 0.2
0.4
0.8
0.8
1.0
Center-Sodium W c e (>rorm)
Elgure 3. Variational XC potential-energycurves for sodium in Cmalong any 5-fold axis. The solid curve is the ground state Cas+ of the positive ion. The similarly shaped curve with long spaces betwecn dashes is the ground state Cm-Na+of the neutral molecule. The very different curve, with its equilibrium position at the center, is the intermolecular chargetransfer state Cm"NaO.The true displacement in energy between these three curvea must also include the quantities given in Table 11. (As explained in the text, the orbital basis set contraction scheme used in this
calculation is slightly different from that used to generate Figure 2). endohedral alkali ion as computed by imposing the three different symmetries studied in this work. The results for lithium, sodium, and potassium are displayed in Table I. The LCGTO basis sets are identical in these three symmetries; thus in the absence of spontaneous symmetry breaking?' which does not occur here, the threedifferent p i b l e calculations for each ion should be identical. The reason that they do not agree is that the charge density and the XC energy density are expanded in incomplete sets of functions. These auxiliary basis sets become more complete in the sequence of symmetries Zh to C5, to C3".Thus the variational principle on the fit to the densitf* requires that the total energy rise in this sequence, which it does not. Thus the error due to an incomplete sampling of the XC energy density is at least as large as the variability in Table I. This variability is reduced by about 10% from the nonvariational calculations reported in ref 30. Another way to check the precision of these calculationswithout significantly increasing their cost is to contract the core parts of the sodium atom orbital basis set according to an atomic calculation on the positive ion, instead of on the neutral. That is done in Figure 3 in calculations along the 5-fold axis. The well depths in Figure 1 are 0.12 eV, and the minima in Figure 1 occur at 0.6 and 0.7 A. Along the 3-fold axis in Figure 2, the well depth of the CmNa+curve is 0.1 1 eV at 0.7 A. Along the 5-fold axis in Figure 3 the well depth of the CmNa+curve is 0.14 eV at 0.8 A. These studies are consistent with CmNa+being a sodium ion inside neutral Cm and the endohedral potential being spherically symmetric. For the PZ LDF the ion is likely to be displaced outward from the center between 0.7 and 0.8 A with an uncertainty of about 0.2 A. The PZ well depth is likely to be 0.15 eV with an uncertainty of about 0.05 eV. We expect similar uncertainties in our calculations of the relative energies for other endohedral alkali complexes discussed herein.
IV. Potential Energy Surfaces Alkali ions M+are closed-shell species electronically and have a rotatonally invariant electronic structure. Thus the carbon cage need not distort to satisfy the Jahn-Teller theorem when the ion is at the center of the Cb0cage. Most excited electronic states
9784 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
Dunlap et al. C,,Li
TABLE 11: First (Ground) .ad Iab.mdceuE;rr Clwge-Trcmsfer(CI’) LDF ASCF Vertical Electroll Affinitie of tbe Cationic Endobedral Comdexes”
neutral electronic state ground (M+Cao-) CT (MOCL)
lithium
sodium
Dotassium
6.05 4.87
6.04 4.70
6.05 3.21
(3-Fold Axis) LDF
“The energies are given in electronvolts and are for the ions at the center of the LDF-optimized Ca cage. of C@M+,however, would require cage distortion if the alkali atom occupied the central position. The lowest lfig example that does not require distortion would involve charge transfer from an aB orbital of Cso to the valence ns orbital of the alkali atom. Because the electron a f f i t y of an alkali cation is so low and the ionization potential of Cm is so high, charge transfer to the endohedral cation requires more energy in the ionic molecule than in the neutral molecule. The ground states of neutral alkali atoms, while not closed-shell electronically, are also rotationally invariant. Thus at least one electronic configuration of C60M will be invariant under the icosahedral group if the alkali-metal atom lies at the center of icosahedral Cas That electronic configuration will not subject the c 6 0 shell to Jahn-Teller distortion. The potential energy surface of this first excited state is compared with the potential energy surface of the ground state (in Figures 2 and 3). The symmetry of this fmt excited state is ABwhen the endohedral atom occupies the central position, and the symmetry of the ground state is Tuwhen the alkali atom is centered. The ground-state surface of the neutral complex is indicated by the dashed line with large gaps and the excited state of this complex is indicated by a dashed line with small gaps. To fit on the same graphs as the ground-state surface of C60Na+,these two states of C60Na are artificially superposed when the alkali atom lies at the center. The LDF energy separations between all three of these different electronic states for each of the lithium, sodium, and pbtassium complexes are given in Table 11. These energies are the difference of separate self-consistent calculations (ASCF). The continuations of these states drawn in Figures 2 and 3 use an LDF approximation. The vertical LDF electron affinities of any closed-shell ion are exactly56
where c(n,R) is the occupation-number-dependent and displacement-dependent oneelectron eigenvalue of the orbital outside the closed shell that is to get the added electron. For nonicosahedral geometries this value is approximated X1e(n,R) dn = c(0,R)
(2)
We expect that the effect of this approximation is less than the effect of not optimizing the c 6 0 cage. The ground state of the neutral endohedral complex along the 3-fold axis splits into A, and E,states, which are degenerate under the approximation of eq 2. Along the 5-fold axis it splits into AI and E states. Consistent with the ground state of the neutral complex being almost purely Cm-Na+, the surface for this state, drawn with the dashed line that has larger gaps in Figures 2 and 3, parallels almost exactly the cationic ground state. Other evidence of this Ca-M+ electronic structure from Table I1 is the fact that the vertical ionization potential of this state is 6.04-6.05 eV for all three alkali atoms when centered. The potential energy surface of the first excited state of the neutral sodium endohedral complex is plotted in Figures 2 and 3 as dashed lines having small gaps. This surface is completely different from the ground state. For it, the equilibrium sodium position is at the center of the Cso shell. (Along both directions the computed relative energy at 0.2 A is downhill by an extremely small amount, 0.01 eV, from the energy at the origin.) From Table I1 this state lies 1.34 eV above the triply degenerate ground state when sodium is at the center, where the transition between these
00
04
08
12
16
20
Center-Lthium Distance (Angstroms)
Figure 4. Variational XC potential-energy curves for lithium in Cso!ong any 3-fold axis. The solid curve is the ground state C&+ of the p a v e ion. The similarly shaped curve with long spaces between dashes is the ground state Cso-Li+of the neutral molecule. The very differentcurve, with ita equilibrium position at the center, is the intermolecular chargetransfer state CsooLio.The true displacement in energy between thme curves must also include the quantities given in Table 11. C,,L
00
04
(5-Fold
08
12
AXIS)
16
LDF
20
Center-lithium mmbce (Angutronu)
Rgwe 5. Variational XC potential-energycurves for lithium in Ca along any 5-fold axis. The solid curve is the ground state C&i+ of the positive ion. The similarly shaped curve, with long spaces between dashes, is the ground state Cso-Li+of the neutral molecule. The very different cwe, with its equilibrium position at the center, is the intermolecular chargetransfer state CaoLio. The true displacement in energy between these curves must also include the quantities given in Table 11. two states is electric dipole allowed. Figures 4 and 5 display corresponding potential energy surfaces for CaLi+ and CmLi with a step size of 0.2 A. The well depths for an endohedral lithium ion are 0.51 and 0.57 eV for the cationic and neutral complexes, respectively, along the 5-fold axis and 0.5 1 and 0.56 eV for the cationic and neutral complexes, respectively, along the 3-fold axis. The minima of both electronic states occur at 1.4 and 1.2 A along the 5-fold and 3-fold axes, respectively. The well is twice as far from the center for lithium as it was for sodium, and it is about four times as deep. As was the case for sodium these potential energy surfaces are spherically symmetric to a good approximation. Likewise, near the origin the CaoLio surface is similar to the C6o”Nao surface, only with an exaggerated energy scale. The dramatic difference between these two potential energy surfaces beyond 0.8 A is due to this difference in energy scale. At this radius the C@%iosurface has risen about 0.5 eV from its energy when the lithium atom is at the origin. At this elevated energy, this surface experiences an avoided crossing with a state having the same symmetry but having a different (attractive) lithium-shell interaction. Compared to the endohedral lithium and sodium complexes, the potassium endohedral complex is quite uninteresting. The potential energy curves for C&+ and C -K+with the potassium ion stepped along the 5-fold axis in 0.2-f increments are plotted in Figure 6. The potassium ion on both surfaces is in equilibrium when it is at the center. We did not attempt to examine a Cs0’%O state beyond the calculations of Table 11, because the first candidate for such a state is the fifth excited state in the icosahedral
The Journal of Physicul Chemistry, Vo1.96, No. 24, 1992 9785
Cm and Endohedral Alkali Atom Interactions C,,K
1.6
(5-Fold Axis) LDF
1
TABLE W. Fitting Pnrrwters for the Potential Energies of Id+ and N d in I ~ m Ca l
/
rot A D, eV a, A-l
I
d2
0.0
p
-O5I
00
hwo, cm-l Be, cm-I d
i
,
,
,
,
,
04
0.3
12
1.3
20
Figure 6. Variational XC potential-energy curves for potassium in Cm along any 5-fold axis. The solid curve is the ground state CaK+ of the positive ion. The similarly shaped cuwe, with long spaces between dashes, is the ground state Ca-K+ of the neutral molecule. The true displacement in energy bctween these curves must also include the electron affinity (ground) given in Table 11.
TABLE IIk LDF Binding Eaergies (io Electronvolts) of the Positively Charged Icosrihednl Endohedrrrl Complex Relative to -rated C, and the Correspoading Pmitively Charged Alkali Iono
CmNa+
CmK+
1.17
1.16
1.16
,I The formations of these complexes are all exothermic, configuration, which we take as an indication that the neutral potassium atom does not really fit inside Cm. If it does in some sense fit, then its size in addition to analogy with this state for lithium and sodium would lead us to expect its potential energy surface to be at least as steeply repulsive between the carbon shell and the alkali atom as the two curves plotted in Figure 6. Table I11 gives the binding of these three CmM+ complexes when the metal ion is at the center relative to separated Cm and M+. All three of these complexes are bound by almost exactly the same amount, which also supports the view that the charge resides on the central atom in each of these complexes. The Cm electronic cloud is radially polarized upon insertion of an alkali cation, however. This last result can be seen by considering the energy-4.1 e V - o f a charge of -e at the center of a sphere of radius 3.5 A and charge +e. For lithium and sodium this binding energy increases 0.1 and 0.5 eV, respectively, when these two ions are displaced 0.7 and 1.4 & respectively, from the center. MNDO predicts that the binding energy of the CmLi+is -2.16 eV in the icosahedral geometry, -1.21 eV from a symmetry-unconstrained optimization with a lithium atom below a hexgonal ring. Thus, the formation of the lithium complex is exothermic in LDF (and ab initioz8)theory and endothermic in MNDO, and the MNDO endohedral well depth, measured from the center, is almost twice as deep as the LDF depth.
V. Fitting the Cationic Potential Energy Surfaces A spherically symmetric model of the motion of an endohedral cation on the potential energy surface is a useful first approximation. Such a description has some similarities with the description of a diatomic molecule. The 5-fold CmNa+and CmLi+ LDF potential energy curves of Figures 1 and 5 have three interesting qualitative features in common: a finite maximum at wo = [ VB”(fo)/p]”2 absolute minimum displaced from the origin; strong repulsion at large r. It is possible to fit these curves by using the functional form” VB = D[1
- P(prO)]z
CaNa+
1.36 0.666 1.54 349 1.31 30.6
0.71 0.254 1.68 130 1.50 31.1
discrete LDF points. First, the curvature at the minimum must be the same. The curvature at the minimum determines the radial vibrational energies within the harmonic oscillator approximation, according to the formula
Center-Potassium Distance (Angstroms)
CMLi+
CmLi+
E”ib
= hwo(v
+ 1/2)
( v = 0, 1 , 2, ...)
(4)
where v is the vibrational quantum number, wo = [V{(ro)/p]1/2, and cc is the reduced mass. Second, the position of the minimum must be the same. The position of the minimum ro determines the rotational energies within the rigid-rotator approximation Ero,=Bjo’+l) 0 ’ = 0 , 1 , 2 )...) where j is the rotational quantum number, and
(5)
*2
is the rotational constant. The ion is assumed to rotate freely about the center of mass of the ion-cage complex. The rotational constant of the cage itself is much smaller than for the ion. The third condition should account, to some extent, for anharmonicity, centrifugal distortion, and rotation-vibration interaction, as in the description of diatomic molecules. Anharmonicity is implicit in the form of VB, as it is for the Morse potential. Furthermore, the unusual feature of the endohedral ion potential, which should be investigated, is the displaced minimum next to a central maximum. Therefore, the third condition is that the difference. between the maximum and the minimum be fit correctly. The position of the minimum, ro, was determined first from a polynomial fit through the discrete LDF points. Then, D and 4 were related to the curvature of the fit at the minimum and the difference between the maximum at the origin and the minimum at ro by the equations VB”(p0)
2a2D
(7)
and respectively. The left-hand sides of the above equations were determined by the polynomial fit through the discrete LDF points, and D and u were determined by a simple iteration scheme. The parameters that were obtained for C60Li+and CmNa+,and the corresponding rotational constants, are given in Table IV. Good fits to the LDF points are obtained in both cases. The Na+ fit is compared to the original data in Figure 7. SchriSdinger’s equation for the radial motion of an ion in the potential VB can be solved analytically for j = 0.” The solution is similar to the Morse potential, except for the boundary conditions. The energy eigenvalues are found numerically from the roots of a transcendental equation involving confluent hypergeometric functions. Numerical experimentation with MATHEMATICAS8has shown that, for the lowest vibrational states, the energies are given by the same formula as for the Morse potential
(3)
where a, D, and ro are positive adjustable parameters. VBwould be the same as the Morse potential if a were negative. VB does not have a horizontal slope at the origin, as the physical curves must have, but this should not be a serious deficiency. The three parameters were determined by requiring some features of V, to be the same as a polynomial fit through the
where d = 2D/hwo = (2pD)’f2/ah is a dimensionless constant. The values for d are also given in Table IV. The s-type radial energy levels of the fitted potential for Na+ are given as horizontal linea in Figure 7. The energy levels for both endohedral Li+ and Na+ are given in Table V. For small quantum numbers v the
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
9786
C,oNet
00
02
04
in the few wavenumber region due to rotations and in the hundred wavenumbers region due to radial well excitations were seen in another model of the interactions of
[email protected] the carbon shell is approximated as a rigid and spherically symmetric shell, there is another rotational constant, roughly 600 times smaller than Be, due to the rotations just of the shell itself.
(5-Fold Axis)
08
OB
IO
12
Center-Sodium Dabnce (Ang8tmmr)
FQwe 7. Threeparameter fit (solid line) to the S-fold axis LCGTO LDF data (dashed line) of Figure 1 using the functional form given in eq 8. The horizontal lines give the first 15 j = 0 vibrational energies of the fitted potential. TABLE V The Twenty Lowest-Energy s-Wave Vibratiod Energies ped Their First Difference in Units of em-' for Endobedral Lithium and Sodium Ions in Buckminsterfullerene LitCm
NatCm
Y
E"
E. - E-I
E"
1 2 3 4
174 514 843 1160 1466 1760 2042 2313 2573 2821 3058 3283 3496 3699 3894 4090 4299 4525 4769 5028
340.1 328.6 317.1 305.6 294.1 282.6 271.1 259.5 248.0 236.5 225.0 213.6 202.6 194.8 196.4 209.0 226.4 243.4 258.9
65 193 316 435 550 66 1 768 873 98 1 1098 1226 1363 1509 1663 1825 1995 2171 2355 2544 2740
5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20
Dunlap et al.
E. - E,, 127.6 123.4 119.1 114.9 110.7 106.9 105.1 108.5 116.9 127.1 137.0 146.1 154.4 162.2 169.6 176.6 183.3 189.7 195.9
anharmonicity is such that the spacing between adjacent energy levels decreases as Y increases. However, when the l e b t turning point is sufficiently close to (or equal to) the origin, the energy differences start to increase with increasing Y. This qualitative change in the level spacing occurs near Y = 16 and 9, for Li+ and Na+, respectively. A fitted potential having a horizontal slope at the origin would give a more pronounced distinction between states for which the origin is classicaly forbidden and states for which the origin is classicaly allowed. The dimensionless constant d is practically the same for endohedral Li+ and Na+ despite a 3-fold increase in mass and a slight increase in a. The question naturally arises as to whether or not this is significant. A possible conjecture is that D and therefore hwo are inversely proportional to the reduced mass p. Extrapolating this relationship to endohedral K+ would predict hoo = 40 cm-I. Assuming an approximately harmonic oscillator potential near a minimum, the ground-state energy would be approximately 20 cm-', and the turning points would be 0.2 A apart. If endohedral K+ moved on a potential energy surface with a displaced minimum, similar to Li+ and Na+, the necessary space for the central maximum and large repulsion might be closer to 0.4 A. The large ionic radius of K+ (1.33 A) would seem to preclude such a potential surface. The approximate molecular rotational constants, Be, for lithium and sodium are both between one and two wavenumbers. This near equality is due to the fact that the equilibrium displacement from the center for Li+ is approximately twice that of the a p proximately three times more massive Na+ ion. Spectral features
VI. Conclusions All-electron local-density-functional (LDF) total-energy calculations were used to study the interaction between icosahedral c 6 0 and endohedral lithium, sodium, and potassium ions and atoms. LDF potential energies as functions of radial displacement from the center along the 5-fold and 3-fold axes of Cm are consistent with the shellalkali interaction being essentially sphericall symmetric. Lithium in its cationic complex is attracted 1.4 from the center toward the Csoshell to gain half an electronvolt in energy. Sodium in its cationic complex is attracted 0.7 A from the center toward the Csoshell to gain a seventh of an electronvolt in energy. Potassium in its cationic complex has its equilibrium position at the center of the c 6 0 shell. The neutral complexes have two very different electronic states. The potential surface of the ground state of the neutral complexes looks almost identical to the ground-statepotential surface of the cationic complex. For lithium and sodium the potential surface of the fmt excited state of the neutral has its minimum very close to the center. The transition between these two different types of electronic states is electric dipole allowed. The existence of a potential energy maximum for the endohedral atom to occupy the central position manifests itself as a minimum in the spacing between j = 0 vibrational energy levels. The height of the maximum is approximately the corresponding vibrational energy. The fundamental vibrational spectrum of endohedral lithium and sodium in CMdivides into four separate regions. Above 200 cm-l are the well-known on-ball vibration^.^"' At about 350 and 130 cm-'we expect the endohedral vibrations of the lithium and sodium ions, respectively. At a few wavenumbers we expect the rotational levels of the molecule as a whole. At much smaller energies we expect the smooth or slightly ratcheted rotation of the Cboshell about its center.
K
Acknowledgment. We thank Rick Mowrey for valuable discussions. This work was supported in part by the US.Office of Naval Research through the Naval Research Laboratory (NRL) and by the NASA Goddard Space Flight Center through Project JOVE.Computations were supported by a grant of computer time from the NRL Research Advisory Committee.
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Ab Initio Molecular Orbital Calculations on DNA Base Pair Radical Ions: Effect of Base Pairing on Proton-Transfer Energies, Electron Affinities, and Ionization Potentials Anny-Wile Colson,+Brent Besler,t and Michael D. Sevilla*-t Department of Chemistry, Oakland University, Rochester, Michigan 48309, and Department of Chemistry, Wayne State University, Detroit, Michigan 48202 (Received: July 6, 1992)
Ab initio molecular orbital calculations have been performed in this study to estimate proton-transfer energies in DNA complementary base pair radical ions and the effect of base pairing on ionization potentials and electron affinities. The calculated (3-21G) adiabatic proton-transfer energy profile for the neutral GC is found to have a single potential minimum, i.e., no stable proton-transfer state, whereas the GC'- shows a second potential minimum which favors proton transfer (AE = -5 kcal). High level calculations (6-31+G(d)) of various uncommon protonation states of the DNA bases and DNA base radical ions were performed to estimate the energy for proton transfer in ' F A , 'C+G, TN-, and CW-. All transfers are energetically unfavorable, but proton transfer in the AT cation radical and anion radical is only slightly endothermic. Base pairing is not found to significantly affect the ionization potential of A or T in the AT base pair. However, base pairing lowers guanine's ionization potential by 0.54 eV while raising cytosine's ionization potential by 0.58 eV. Base pairing revem the order of the ionization potentials and electron affinities of thymine and cytosine which makes cytosine the most electron affinic DNA base and the least likely to be ionized. The order of ionization potentials in base pairs calculated at the 3-21G level is C > T >> A > G. Further investigation was performed on stacked four base (AT/GC) configurations. A 3-21G calculation of the anion radical of the stacked system with the neutral base pair geometries shows the electron localizes on thymine. However, on relaxation of the nuclear framework of the AT/GC system, the electron is found to preferentially localize on cytosine. Calculations on the effect of ion formation on hydrogen bond strengths in base pairs suggest that the hydrogen bonds in the GC anion and cation are greatly strengthened over the neutral GC parent, whereas only the hydrogen bonds in the cation are strengthened in AT over the neutral parent.
Introduction Initial localization of charge in irradiated DNA has been the focus of a number of studies which s& to aid our understanding of primary radiation damage processes in D N A . I - ~ Experiments Oakland University. *Wayne State University.
0022-365419212096-9781$03.00/0
on yirradiated DNA at low temperature have shown that the electron 10CdiZes preferentially on the cytosine base in doublestranded DNA and thymine in single-stranded DNA whereas the hole localizes On guanine in both forms.' The initial localization of the electron and hole on the DNA will depend largely on the electron affinities and ionization potentials, respectively, of the individual DNA bases. It is clear from a number of recent studies Q 1992 American Chemical Society
