J. Phys. Chem. 1988, 92, 21 15-2120 TABLE VII: Electronic States and Spatial Configurations for MO'
bond energy, bond kcal/mol order s c o + 159 I 3" TiO+ 161 5 3" vo+ 131 f 5 3" CrO+ 8 5 f 1 2II2 MnO+ 5 7 * 3 2b FeO' 6 9 f 3 2b coo+ 64 3 2' NiO' 45 f 3 1112
*
*
*
"Triple bond as in VO'. bonds as in RuO+.
sld"' M+ MO' confign
dn M+ MO+
confign
bond 6 svm order u T 6s 2 22 00 l2' 2 22 10 2A 2 2 2 11 s22 32 11 4JJ 2 3 3 11 '2' 21/2 2 3 2 11 1 2 33 21 4A 2b 2 33 11 1 2 33 22 ' 2 - 2b 2 33 21 1 u
T
2 43 22
2n
26
2 33 22 1
svm
'II 6Z+
'A 42-
b u bond plus the two three-electron
T
relevant. This does not lead to different predictions for configurations for the early metals but leads to a second set of possible ground configurations for the late metals, as indicated in the second column of Table VII. Special points to note are as follows: (a) ScO', TiO', and VO+ can form strong triple bonds (with 0, 1, or 2 electrons in 6 orbitals) with symmetries IZ+,,A, and 32-,respectively, but CrO+ (411) necessarily has an electron in an antibonding orbital, leading to much weaker bonding. (b) For MnO', one would expect a molecular configuration related to s'd5 Mn+ and hence a 511 state. Close in energy should be a 52+state, derived from the d6 configuration of Mn+, forming an O,-type bond to oxygen atom. (c) The ground state of FeO+ is expected to differ from that found for RuO+ in section 111. Instead of the 'A ground state found for RuO+, we expect a 62+ground state for FeO+. This change in electronic states may be understood simply in terms
Anisotropic Polarizability Density in the H,'
2115
of the change in ground state of the metal ions, with the s1d6 ground state of Fe+ and the d7 ground state of Ru+ leading to 0,-type bonds of %+and 'A symmetries, respectively (the nonbonding d electrons on Fe+ having a preferred configuration of a'6I8', while the s and three d n electrons comprise the 02-type bond). symmetry (d) COO' should have a low-lying excited state of and a ground state of 32-symmetry, arising from 0,-type bonds to oxygen to the s1d7excited state and the d8 ground state of Co+. The 5A excited state of COO' has an extra 6 electron compared with Fe0+(?3+), while the 3Z-ground state of COO' has both 6 orbitals doubly occupied. (e) NiO' is predicted to have a ,II ground state involving the d9 ground state of Ni' forming a single u bond to oxygen (with an unpaired A electron left on 0),with a 42-excited state derived from an 0,-type bond to the s1d8excited state of Ni'. The low bond energy of NiOf (45 kcal/mol) is consistent with the formation of a single bond as in 211 NiO'. (0 As a final note, ligand field effects in porphyrins should yield d" ground states for M2+. Thus the electronic states of M02+in porphyrins should correspond to the lowest electronic state predicted for d" MO', namely, a 4A ground state for FeO', while COO' (porph) and NiO+ (porph) should have 3Z-and ,II ground states, respectively.
Acknowledgment. This work was supported by the National Science Foundation (Grant No. CHE83-18041) with some support from the Shell Companies Foundation. E.A.C. acknowledges a National Science Foundation predoctoral fellowship (1982-1985), a research grant award from the International Precious Metals Institute and Gemini Industries (1985-1986), and a S O H 1 0 fellowship in Catalysis (1986-1987). Registry No. VO', 12192-26-6;RuO', 113110-48-8.
Molecule
David H. Drum and William H. Orttung* Department of Chemistry, University of California, Riverside, California 92521 (Received: July 9, 1987)
Formal anisotropic polarizability densities were obtained from quantum charge increments in a small uniform applied field by integrating the general mathematical relation between a charge increment and the divergence of a charge shift polarization. A Gaussian basis set was developed to evaluate the field-induced quantum charge increments. The results for parallel and perpendicular applied fields were fitted to relatively simple Legendre-Laguerre expansions in the elliptic coordinates. Internuclear separation was varied between 1.4 and 3.0 au. The unusual features of H2+relative to H2 and to diatomics with inner-shell electrons are discussed.
Introduction The H2+ molecule was chosen for initial studies of the distributed anisotropic polarizability of a chemical bond primarily because it is a one-electron system. The time-averaged charge increment at each point depends only on the nature of the electron's motion in the nuclear and applied fields. Coulomb repulsion from other parts of the electron cloud is not a consideration because an electron is essentially a point charge on an atomic scale and does not interact with itself at spatially separated points. In a formal (mathematical) sense, A p , the field-induced quantum charge increment at each point of space, can be interpreted as the negative divergence of a polarization P: -V.P = Ap (1)
P(r) represents the dipole moment density of the shifted charge density at point relative to its zero-field position, and eq 1 expresses only conservation of charge: If the charge shift varies across a local region, there will be a net local charge increment. 0022-3654/88/2092-2115%01.50/0
The charge density exists in a time-averaged sense at each point of the molecule (with or without a field), and is evaluated by solving the Schroedinger equation. While the above relation is independent of dielectric theory, the close analogy to it suggests that P may be expressed as the product of a formal polarizability density tensor X and a small local field E. (This is the applied field in the case of a one-electron system.) Our interest in the continuum dielectric formalism as a model for atomic interiors was stimulated by the work of Oxtoby and Gelbart.' A number of other approaches have been e ~ p l o r e d . ~ - ~ (1) (a) Oxtoby, D. W.; Gelbart, W. M. Mol. Phys. 1975, 29, 1569. (b) Mol. Phys. 1975, 30, 535. See also: (c) Theimer, 0.;Paul, R. J . Chem. Phys. 1965, 42, 2508. (d) Frisch, H. L.; McKenna, J. Phys. Rev. A 1965, 139, 68. (2) (a) Heller, D. F.; Harris, R. A.; Gelbart, W. M. J . Chem. Phys. 1975, 62, 1947. (b) Gina, J. A.; Harris, R. A. J . Chem. Phys. 1984, 80, 329. (3) (a) Applequist, J. Acc. Chem. Res. 1977, 10, 79. (b) Chem. Phys. 1984, 85, 279, (c) J , them, phys, 1985, 83, (4) Oxtoby, D. W. J . Chem. Phys. 1978, 69, 1184.
0 1988 American Chemical Society
2116
The Journal of'Physica1 Chemistry, Vol. 92, No. 8, 1988
The idea of a formal polarizability density arises from the classical continuum dielectric model of an atomic interior and does not appear to be derivable as a local property from quantum mechanics. Indeed, it has been shown that, in the case of simple one-electron model systems, (including the hydrogen atom),I0 X is not a purely local function (as it is in a classical dielectric) since the value deduced for the function depends on the spatial variation of the field. Nevertheless, we feel that this function provides an interesting and potentially useful description of the response of the electron cloud to an applied field. The description should become more legitimate as the number of electrons and atoms in a molecule increases. The polarization can be evaluated from quantum charge increments in a uniform applied field if eq 1 can be solved for P. P is then a quantum result in the sense that it does not depend on the models usually associated with dielectric theory. In the general case, the vector nature of eq 1 makes it difficult to solve, and the solution (if found) does not specify the rotational part of the polarization. The latter quantity is closely related to assumptions made about the anisotropy of X(r), the polarizability density tensor. (It is hoped to discuss this interesting topic in a future publication." In the present paper, the rotational part of P is not explicitly considered.) It has been shownI2 that eq 1 can be integrated in an average sense for a linear molecule in a field parallel to the axis. In this approach, eq 1 is first reduced to a one-dimensional equation by integrating over planes perpendicular to the internuclear axis. The resulting one-dimensional radially averaged polarizability density provides an easily comprehended description of the response of the electron cloud to an axial applied field. As demonstrated in the Methods section below, this "cylinder integral" method may be generalized to cover perpendicular applied fields to give a one-dimensional radially averaged perpendicular polarization. In the special case of H2+,if the polarizability tensor is assumed to be diagonal in a local axis system aligned with the cylindrical coordinate increments, then eq 1 can be integrated directly to yield three-dimensional anisotropic polarizability densities. The polarizability densities evaluated from the quantum charge increments can then be fitted to relatively simple analytic functions. It is the purpose of this paper to present results of this type for
H2'. It is hoped that the study of H2+will assist the investigation of more complex many-electron bonds, even though H2+is unusual in several respects. H2+ (unlike H2) has an unbounded parallel polarizability as the nuclei move to larger separation. In addition, H2+(like H2) differs from heavier diatomics in its lack of filled inner shells, and both are more like a united atom than a typical diatomic molecule at the equilibrium internuclear distance. The literature on H2+is extensive, and we mention only those references that are directly relevant to the present work. Accurate energiesI3.l4 and wave f u n c t i o n ~ l allowed ~ . ~ ~ us to check our quantum calculations with Gaussian basis sets. We are not aware of earlier use of Gaussian basis sets for H2+ in the literature, although we have benefited from work on the hydrogen molec~le.'~ A number of authors have made precise quantum mechanical calculations of the total (or global) polarizability components of (5) Munn, R . W.; Luty, T. Chem. Phys. 1983, 81, 41. (6) (a) Hunt, K. L. C. J . Chem. Phys. 1983, 78,6149. ( b ) J . Chem. Phys. 1984, 80, 393. (7) Pearson, E . W.; Waldman, M.; Gordon, R. G. J . Chem. Phys. 1984, 80, 1543. ( 8 ) (a) Soven, P. J. Chem. Phys. 1985,82,3289. (b) Bobel, G.; Longinotti, A.; Fumi, F. G . J . Phys. (Les. Ulis, Fr.) 1987, 48, 45. (9) Stone, A. J. Mol. Phys. 1985, 56, 1065. (10) Orttung, W. H.; Vosooghi, D . J . Phys. Chem. 1983, 87, 1432. (11) Drum, D. H.; Orttung, W. H., manuscript in preparation. (12) Orttung, W. H. J . Phys. Chem. 1985, 89, 3011. (13) Peek, J. M. J. Chem. Phys. 1965, 43, 3004. (14) Bishop, D. M.; Wetmore, R. W. Mol. Phys. 1973, 26, 145. (15) Bates, D. R.; Ledsham, K.; Stewart, A. L. Philos. Trans. R. SOC. London A 1953, No. 246, 215. (16) Roberts, E. M.; Foster, M. R.; Selig, F.F. J . Chem. Phys. 1962, 37, 485. (17) Dykstra, C. E. J . Chem. Phys. 1985. 82, 4120
Drum and Orttung
H2+.18-20Some of the earlier work has also been summarized.2' Accurate global hyperpolarizabilities have also been calculated.22 Limitations of the finite-field method occur only at much higher field strengths and for more extensive basis sets than we have used.23 The importance of vibrational contributions to polarizability and hyperpolarizability has recently been i n ~ e s t i g a t e d . ~ ~
Methods To apply the "cylinder integral" method for linear molecules,'2 the charge increments are first integrated over planes perpendicular to the molecular axis. In cylindrical coordinates, z is along the axis, r is perpendicular to it, and 4 increases around the axis from x toward y. (The assumption of a diagonal X tensor is not made in this case.) Consider the case of a parallel applied field first. If the quantum charge increment App(z,r) is integrated over r a n d 4 to obtain a one-dimensional increment +,(I), then the onedimensional analogue of eq 1 is -dPp(')/dz = App(')
(2)
which may be integrated directly to give
(3) where PP(') is zero at large z. For a perpendicular applied field, the cylinder integral method may be generalized as follows: The quantum charge increment, Aps, has a Pl(cos 4) component for small fields, with amplitude Apsl in unit field along the 4 = 0 direction (corresponding to the x axis and field direction); thus the induced moment per unit z of a plane perpendicular to z is given by PJ1)(z) = L m L 2 r ( rcos 6)(ApSl cos $ ) r d 4 dr
(4)
or (5)
which is analogous to eq 3. Equations 3 and 5 may be integrated along z to give the total polarizabilities ap and as. Alternatively, the moment of the quantum charge increment may be integrated to give the same total polarizabilities. A third estimate may be obtained from the orbital coefficients as the dipole moment divided by the field strength. (The quantum programs are described in the Calculation section below.) If we define a polarizability density tensor X by the equation
P = XE
(6)
and assume that E is a small uniform applied field, then eq 6 may be substituted in eq 1, which may then be integrated if it is assumed that X is diagonal in a local coordinate system, dz, dr, r d4. The equation to be integrated is dP,/dz
+ (1 /r)
a(rP,)/dr
+ (1 / r ) dP,/d&
= -Ap
(7)
where P, = X,E,, P, = X,E,, P, = X,E,
if X is diagonal in the sense of the preceding paragraph. For the Bishop, D. M.; Cheung, L. M. J . Phys. B 1978, 1 1 , 3133. Montgomery, H . E., Jr. Chem. Phys. Lett. 1978, 56, 307. Aubert-Frecon, M.; Le Sech, C. J . Mol. Spectrosc. 1981, 87, 56. Murai, T.;Takatsu, H. J . Phys. SOC.Jpn. 1978, 45, 1704. Bishop, D. M.; Cheung, L. M. J. Phys. B 1979, 12, 3135. Bishop, D. M.; Solunac, S. A. Chem. Phys. Lett. 1985, 122, 567. (a) Bishop, D. M.; Solunac, S. A. Phys. Reu. Lett. 1985, 55, 1986. (b) Adamowicz, L.; Bartlett, R . J . J . Chem. Phys. 1986, 84, 4988. (18) (19) (20) (21) (22) (23) (24)
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988 2117
Anisotropic Polarizability Density in H2+ special case of unit field E , parallel to z, eq 7 reduces to ax,/az = - ( A ~ ) ,
(9)
For unit field E, parallel to x
P, = X , cos 4, P, = -X, sin 4
(10)
and eq 7 becomes
ax,/&cos 4 + ( X , / r ) cos 4 - ( X , / r ) cos 4 = -(Ap),
(1 1)
If (Ap), is expanded in a Legendre polynomial series
and substituted in eq 10, then only the PI term is retained in small fields, and
If X , = X , (an assumption similar to that made for atomsI0."), then
axr/ar = -(A~)sl
(14)
Equations 9 and 14 may be integrated from large z or r (where Ap = 0 and X = 0) to the z or r value of interest, holding the other coordinate ( r or z ) constant:
X,(z,r) = S - ( A p ) , dz',
X,(z,r) = I m ( A p ) s ldr' (15a, 15b)
Several of the above results have assumed that the X tensor is diagonal in a local z,r,@cylindrical coordinate system, or that
"The basis functions for the atoms and origin were developed as follows: The atom S functions 2-11 are the Is-10 GTO basis of Hu~inaga.~' Function 1 is 8 times function 2, and functions 12-14 are successively reduced from function 11 by factors of 3-4. The atom P functions 1-6 are the 2p-6 GTO basis of Hu~inaga.~'Functions 7-8 are successively reduced from function 6 by factors of 3.0 and 3.5. The atom D functions 1-3 evolved from the 0.65, 0.1 GTO pair of Dykstral' for H,. The origin S functions were the same as 8-10 of the atom S functions, and the origin P function was the same as function 3 of the atom P set. *Geometric scale factors vs R are discussed in the text. The uncontracted Gaussian basis set described in Table I was found to be adequate for our purposes. It was adjusted to improve both energy and polarizability. The following geometric scale factors, s, were applied to all functions of the basis at the four internuclear separations, R, used in the calculations:
R (au) S
1.4 1.77
2.0 1.52
2.5 1.39
3.0 1.29
s values are qualitatively similar to those of Coulson.28 The first
Xzs and Xrr are zero by symmetry, but X , is not. In the case of the hydrogen atom,I0 it was assumed that X,,= X,, (in spherical polar coordinates), Le., that there was no radial-angular anisotropy. If H2+ is approximated by the additive combination, (H H)/2, then it is easily shown that the contribution to X,, from one of the hydrogen atoms is
+
e sin e
exponent type atoms 9360.000, 1170.498, 173.5822, 38.65163, 10.60720, 3.379649, S 1.202518, 0.463925, 0.190537, 0.0812406, 0.0285649, 0.008542, 0.0022221, 0.0005115 3.00971 1, 0.710128, 0.227763, 0.085676, 0.035652, 0.015442, P 0.005 1473, 0.0014707 D 0.591044, 0.139979, 0.043283 origin S 0.463925. 0.190537. 0.0812406 P 0.227763
(Note that s > 1 implies a compression of the function.) The above
x,, = x,, = x,,= 0
X,, = '/,(Xrr- x,,) cos
TABLE I: Uncontracted and Unscaled Gaussian Basis Set for the Hydrogen Molecule Io&
(17)
so that X,, would also be zero as a result of the assumed atomic isotropy at this crude level of representation of H2+. (Note that the subscripts on the right side of eq 17 refer to polar coordinates.) The assumptions concerning anisotropy of the polarizability density affect the solution of eq 1 by varying the rotational part of the solution for P (the part for which V X P # 0). We believe that our assumptions ( X , = X , in atoms, and X,, = 0 in diatoms) are appropriate because of their simplicity and because the results are similar to those expected for a continuum dielectric. The subtleties of the rotational part of the polarization vector will be considered in detail elsewhere.11
Calculations The quantum calculations were carried out on a VAX 11/785 using the HONDO program system,25 modified to allow small uniform applied electric fields.'z-26 Charge densities were calculated from orbital coefficients by our own programs. Nonlinear field effects were detected as unequal charge increment magnitudes up and down field relative to the center of the molecule. These effects could be seen in the less significant digits (for parallel applied fields) even for field strengths down to 10" or lo-' au. All of the charge increments involved in this work are differences au and zero field strengths. of calculations at (25) Dupuis, M.; Wendoloski,J. J.; Spangler, D. Not[. Resour. Comput. Chem. Software Cat. 1980, 1 , QGO1. (26) An error in the elimination of eigenvectors with small eigenvaluesin the overlap matrix was corrected, and the two-electronintegrals were left out of the H2+calculations. (27) Huzinaga, S. J . Chem. Phys. 1965, 42, 1293.
two R values correspond to the equilibrium separations in H2 and H2+, respectively. To obtain polarizability densities from the quantum charge increments (adjusted to unit field strength), the charge increments were calculated for the points of a two-dimensional rectangular array in the z,r plane. The polarizabilities for parallel and perpendicular fields were evaluated from these charge increments according to eq 15. The simplex method was then used to vary the parameters of trial functions to obtain the best least-squares fit to the array of points. The modified simplex was used initially and was then replaced by the supermodified simplex method." The response at the centroid was estimated.j2 Each fit had linear amplitude and nonlinear exponential parameters. A large reduction in the number of iterations was achieved by mixing the simplex method and multiple linear regression. In each iteration cycle, the new set of nonlinear parameters was determined by the simplex method and the linear parameters were then fit by regression. The problem of degeneracy in the simplex should be noted.33 Each fit was restarted until stability was achieved. All of the reported fits represent stable minima, though not necessarily global minima. Several types of analytic functions were tried in the effort to fit the polarizability data. Initially, generalized hydrogen atom functions34were placed at the nuclei. When functions were added at the center of the molcule, there was a significant improvement (reflecting the united atom character of Hz+). Distortion functions were added at the nuclei. Good fits of the perpendicular polarizability were obtained with these functions, but the results for (28) Coulson, C. A. Trans. Faraday SOC.1937, 33, 1479. (29) Nelder, J. A.; Mead, R. Comput. J . 1965, 7, 308. (30) Caceci, M. S.; Cacheris, W. P. BYTE 1984, 9, 340. (31) Routh, M. W.; Swartz, P. A,; Denton, M. B. Anal. Chem. 1977,49, 1422. (32) Van der Wiel, P. F. A. Anal. Chim. Acta 1980, 122, 421. (33) Van der Wid, P. F. A,; Maassen, R.; Kateman, G. Anal. Chim. Acta 1983, 153, 83. (34) Orttung, W. H.; St. Julien, D. J , Phys. Chem. 1983, 87, 1438.
2118
Drum and Orttung
The Journal of Physical Chemistry, Vol. 92, No. 8. 1988
TABLE 11: Global Results of Quantum Calculations: Comparison with Fitted Functions"
total energy (exact)b HONDO
R = 1.4
R = 2.0
R = 2.5
R = 3.0
-0.56998 -0.569979
-0.602634 -0.602632
-0.593824 -0.577563 -0.593821 -0.577561
2.09758 2.09757 2.09722
5.07765 5.07767 5.07660
10.1495 10.1495 10.1476
19.6995 19.6997 19.6943
1.12583 1.12579 1.12543
1.75765 1.75746 1.75726
2.34342 2.34303 2.34237
2.93661 2.93591 2.93504
2.095 1.126
5.066 1.758
10.115
2.343
19.619 2.934
polarizability P
(lit.)c HONDO
N Id S
(lit.)c HONDO
NId
polarizability from fitted functions P S
'All values are in atomic units (au): p and s refer to parallel and perpendicular applied fields, respectively. Bates, Ledsham, StewartI5 at R = 1.4; PeekL3at larger R values. These values are essentially exact within the Born-Oppenheimer approximation. Bishop and Cheung;Is Aubert-Frecon and Le Sach;20 M ~ n t g o m e r y . ' ~ dNumerically integrated quantum charge increments, as described in the text.
2
Figure 1. Quantum charge density for H2+(-) and (H + H)/2 (---). The upper curves are cylinder integrals, and the lower curves (with cusps) are the values along the internuclear axis. The center of the molecule is at the origin, and one nucleus is at R = 1 au.
parallel polarizability were less satisfactory. The greatest deviations occurred in the region between the nuclei. A better fit was obtained by using elliptic coordinates, X = (ra r b ) / Rand p = (r, - r b ) / R ,and expressing X as an expansion of the following type
+
A:
where CJ = As,, s, is a scale factor, PIis a Legendre polynomial, ~ first three Laguerre and L, is a Laguerre p ~ l y n o m i a l . ~The polynomials (Lo, L , , and L 2 ) are 1, 1 - z, and 2 - 4z + z 2 , respectively. (Laguerre polynomials seemed to work better in the simplex fitting process than a power series.) Similar expansions have been used in the quantum calculations for H2+.14 The quantum calculations used products of sums over 1 and q because the Schroedinger equation is separable in X and p. For our purposes, the sum of products was more flexible, since each Legendre polynomial could then have a different radial dependence. This flexibility did lead to a better fit of our data. The perpendicular fit was excellent when at least six combinations were used ( q 112 I2 ) . For the same set, the parallel fits were good, but there were small deviations between the nuclei. An additional Legendre polynomial caused oscillations in the fit, but the addition of an anisotropic Gaussian function at the center of the molecule greatly improved the fit. The form of the Gaussian was
1
I
Table I1 summarizes the global aspects of the calculations. The quantum energies typically agree with literature values to more than five digits (0.001%). The parallel polarizabilities (from HONDO quantum calculations) are in equally good agreement, while the perpendicular polarizabilities have larger errors, and are systematically low. The fitted functions do not give as accurate results as the quantum calculations. The most interesting numerical results of this paper are the polarizability density functions presented in Figure 3. These funtions were deduced from the quantum charge increments. The coefficients of the fitted functions are shown in Table 111. While not as accurate as the quantum results, the fitted functions allow the reader to generate the functions to a few percent reliability in a convenient manner. At R = 2.0 au, the fits are good to about (35) Levine, I. N. Quantum Chemistry, 3rd ed.;Allyn and Bacon: Newton, MA, 1983; pp 133-134.
_ _ _
R- 2 . 0 AU
I
I
I
I
2. 00
0. 00
A.
00
2
I
I
I
I
B:
+
Results
H2*.
Hz*.
I
R= 2 . 0 AU
.
. _ .(H+H)/2
00
1
I
0. 00
I
1
2. 00
I
A. 00
2
+
Figure 2. Quantum charge density increments for H2+ (-) and (H H)/2 (---). The curves in part a (top) are cylinder integrals, and the curves in part b (bottom) are along the internuclear axis. The increments are for unit applied field parallel to the molecular axis.
1% for perpendicular field, and to about 1% for parallel field except between the nuclei, where the fit is about 4% low. The uncertainties are similar but about twice as large at R = 3.0 au. Figure 1 shows several aspects of the quantum charge density of H2+in comparison with that of two half hydrogen atoms, (H + H)/2, at the ground-state equilibrium separation of R = 2.0 au. These results are not new but provide a useful reference for the polarizability density results. The charge density in the molecule is considerably compressed relative to the atoms in both the axial and radial directions. This effect becomes more no-
The Journal of Physical Chemistry, Vol. 92, No. 8, 1988 2119
Anisotropic Polarizability Density in H2+ TABLE 111: Polarizability Density Function Parameters for
coo
R 1.4 2.0 2.5 3.0
coz
co1
3.966 3.255 0.492 -10.23
-2.523 -3.780 -5.187 -8.815
0.2203 0.3375 0.4653 -0.2472
-1.103 13.87
0.5747 2.471 9.367 36.99
H2+in a Uniform Parallel or Perpendicular G o c21 c40 Perpendicular -3.146 -0.6186 -9.092 -1.749 -19.95 -3.682 -41.31 -7.207
Applied FieldD
-0.2047 -1.877 -14.11 -152.7
SO
s2
s4
4.371 5.702 6.699 6.509
3.841 5.051 5.995 6.907
4.396 7.094 10.23 14.41
4.613 6.392 7.969 9.622
4.641 6.436 7.891 9.288
7.076 9.056 10.84 14.49
Parallel 1.4 2.0 2.5 3.0
37.98 216.6 1028 5172
101.8
576.5
-18.04 -126.9 -558.9 -2201
R
a,
-2.573 -15.54 -61.02 -218.4
-3.044 -28.61 -160.1 -2246
c,
“2
Parallel Anisotrooic Gaussian 1.893 3.517 1.288 2.982 1.023 2.710 0.8895 2.520
1.4 2.0 2.5 3.0
0.8970 2.223 4.218 7.780
“See eq 18 and 19 for definitions of the parameters. I
I
1
1
I
I
I
H2*.
Ep.
O
R
=
2.0 AU
OM
m
0
I
i I
\ ‘,
0
L
2
I \
8 : HzL.
R-
2.0 AU
1
0.09
1. 50 Z (AU)
3. 00
Figure 4. Contributions of the L = 0, 2,4, and Gaussian terms of eq 18 and 19 to the parallel field polarization in unit parallel field. The total and quantum results from Figure 3b are also shown. The curves are along the internuclear axis. Between the nuclei (121 < R / 2 ) , h = 1 and varies from -1 to + l . For z > R/2, h > 1 and p = 1.
axis are shown. The charge-transfer effect is very clear in the comparison of H2+ and (H + H ) / 2 . Figure 3 shows the variation of P, and P,, the polarizations per unit fields parallel and perpendicular to the z axis, in comparison with the result for ( H H)/2. Since P = XE,and E = 1, Figure 3 is also a plot of X , and X,. The results in Figure 3 were obtained by integration from results like those of Figure 2, as described under Methods. The cylinder integrals have somewhat different z dependence than the values along the axis. In the (H H ) / 2 case, results are the same for parallel and perpendicular fields since X is an isotropic tensor in this case (because the atomic functions were assumed to be isotropic). Figures 4 and 5 show the contributions (along the axis) to the total fit of X a t the equilibrium internuclear distance. The perpendicular fit is considerably better than the parallel fit. Both fits are excellent when plotted as the cylinder integral.
+
i
Figure 3. Polarization in a unit applied field parallel or perpendicular
to the molecular axis. The curves were obtained by integrating the quantum charge increments parallel or perpendicular to the internuclear axis, as described in the text. The curves in part a (top) are cylinder integrals, and the curves in part b (bottom) are along the internuclear axis. H2+in parallel (-) and perpendicular (---) field, and (H + H)/2 (---). Note that since (H + H)/2 is isotropic (as discussed in the text), the response is the same in parallel and perpendicular fields.
ticeable at shorter internuclear distances. It is interesting to note the difference in shape of the axial values relative to the cylinder integral values as a function of z . This difference reflects the manner in which the density function varies in the radial direction while moving along the axis. Although the values near the axis contribute very little to the volume integral of any property, they are often of interest, in part because the nuclei are on the axis. Figure 2 shows the variation of the quantum charge increments (per unit field) in small au) parallel applied field for H z + and (H H)/2. The cylinder integral and values along the z
+
+
Discussion The most striking feature of the results is the strong anisotropy of the polarizability density in the region between the nuclei. Although an effect of this type might generally be expected for a chemical bond (because of the greater geometric extent of the bond in the parallel direction), the primary explanation in H2+ appears to be the charge transfer that occurs in parallel applied
2120
J . Phys. Chem. 1988, 92, 2120-2124
-
c. 3T
i
1-
~
I. 50 L (4.)
L
~
L
-
3. 0 3
Figure 5. Like Figure 4, but for perpendicular applied field. No Gaussian term was needed for a close fit in this case.
field. Unlike a typical (second row or higher) diatomic, H2 and H2+behave more like a single united atom in a parallel field. For example, the charge increment is an odd function relative to the center of the molecule (rather than to the centers of the bonded
atoms). In a perpendicular applied field, we observe more typical diatomic behavior. The strong increase of the parallel polarizability of Hz+with increasing internuclear distance is a closely related effect. Since the potential has a double minimum, a small applied field causes a larger shift in the average position of the electron at larger internuclear separations. The parallel polarizability of H2+appears to increase without limit with increasing R, in contrast to that of H2, which is more stable with one electron near each nucleus at large R . The parallel polarizability of H2 goes through a maximum with increasing R , and then falls to the value for two independent hydrogen atoms.36 We conclude that, while the one-center and one-electron aspects of H2+may cause it to be atypical relative to diatomics in general, the absence of electron interaction allows a more detailed analysis to be carried out. The same limitations apply to the present results as applied to the earlier calculation on the hydrogen atom.I0 The present results are strictly applicable only to uniform applied fields and reflect the assumption of a diagonal polarizability tensor in local cylindrical coordinates. Until a theory is devised for answering the question of radial/angular anisotropy of the hydrogen atom and the off-diagonal X,, for H2+,the present assumptions seem to be justified as the simplest that allow a solution, and the closest to the standard continuum dielectric formalism.
Acknowledgment. The research was supported by grants from the Academic Senate Committee on Research and from the Academic Computing Center, University of California, Riverside. Registry No. H2+, 12184-90-6. (36) Kolos, W.; Wolniewicz, L. J . Chem. Pbys. 1967, 46, 1426.
Tlme-Resolved ESR Detection of the Phosphorescent States of Xanthone in Glassy Matrices at 77 K Hisao Mumi,+Masashi Minami, and Yasumasa J. I’Haya* Department of Materials Science and Laboratory of Magneto- Electron Physics, The University of Electro-Communications, Chofu, Tokyo 182, Japan (Received: August 4 , 1987; In Final Form: November 11, 1987)
The excited triplet states of xanthone in organic glasses are studied by using a time-resolved ESR method at 77 K. The observed spectra are found to be superpositionsof three kinds of triplet states in polar media. The mixing ratio of the three spectra varies with the change of the solvent. The spectrum that shows a small zero-field splitting parameter D becomes predominant in water-containing solvents and is assigned to a purely isolated 37r7r* state influenced by a strong hydration due to water molecule impurity. The other two spectra are assigned to the %7r* states having large D values, which are strongly coupled through spin-orbit interaction with the 3 7 r ~ *state located a little higher in energy. The existence of these two %T* states in dry polar glasses may be attributed to structurally different conformers in the inhomogeneous circumstance of the glassy condition. In nonpolar glasses, no excited triplet state of xanthone is observed when an X-band ESR spectrometer is used because of its huge D value, but a weak total emissive signal of the excited %T* of hydrated xanthone existing as an impurity is detected.
Introduction Xanthone is one of the few compounds that show a dual phosphorescence phenomenon. Many research groups have investigated this interesting and mysterious problem extensively, but all the data available to date have been obtained under different conditions and are not sufficient to settle the problem. Pownall and co-workers proposed the existence of two kinds of molecules whose ground states are different from one another, +Present address: Department of Chemistry, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan.
0022-3654/88/2092-2120$01.50/0
possibly geometrical conformers in matrices,’ and similar ideas were employed by other groups to explain the complex dual phosphorescence of xanthone.2 On the other hand, assumption of a thermal equilibrium between two triplet states in the molecule, the lowest 37r7r* and 3n7r* states, the latter having a higher energy, is another explanation for this phen~menon.~-~. Thus, there still (1) (a) Pownall, H. J.; Huber, J. R. J . A m . Chem. SOC.1971, 93. 6429. (b) Pownall, H. J.; Connors, R. E.; Huber, J. R. Chem. Phys. Lett. 1973, 22, 403. (c) Pownall, H. J.; Mantulin, W. W. Mol. Phys. 1976, 31, 1393. (2) (a) Connors,R. E.; Walsh, P. S. Chem. Phys. Lett. 1977, 52, 436. (b) Vala, M.; Hurst, J.; Trabjerg, I. Mol Phys. 1981, 43, 1219.
0 1988 American Chemical Society
