J. Phys. Chem. 1991,95,2307-2311 TABLE VI: Atomic Panmeters atom orbital Hi,, eV €1 V 4s -8.81 1.30 4p -5.52 1.30 -11.00 4.75 3d -32.20 2.28 0 2s -14.80 2.28 2p 1.63 C 2s -21.40 2p -I 1.40 1.63 1.30 H IS -13.60
2307
Appendix 52
CI"
C2"
1.70
0.4755
0.7052
"Coefficients used in the double zeta expansion of the metal d-orbitals.
studies of these later steps of the proposed reaction path will definitely be of great interest. Acknowledgmenr. B.S. is thankful to the Carlsberg Foundation for economical support and to Thanks to Scandinavia, Inc. (especially Mr. Richard Netter) for making the stay at Cornel1 possible. We are thankful to Arne Lindahl for his careful p r e p aration of the drawings and to Susan Jansen for sending a paper prior to publication.
The extended HUckel19 molecular and tight bindingm approach is applied throughout this study. Bond distances and angles for the different surface models are taken as average of the experimental values reported for the vanadyl pyrophosphate structure.'* This leads to a V-V distance of 3.188 A and a V=O distance of 1.628 A. The geometry of 2,5-dihydrofuran is in accordance with the experimentally reported structure.30 Butadiene has C-C separations of 1.48 and 1.34 A and all angles are 120°.3' The dioxygen species have an 0-0distance of 1.40 A.M The distance between vanadium and an oxygen from the adsorbed dioxygen is 1.90 A.2s Atomic parameters are listed in Table VI. In the tight-binding calculations, properties averaged over the Brillouin zone were estimated by the use of a 10 k-point set according to the geometrical method by Ramirez and BOhm.'* (30)Courtieu,J.; Gounelle, Y.Mol. Phys. 1974, 28, 161. (31) March, J. Aduanced Organic Chemistry, 3rd ed.; Wiley Interscience: New York, 1985; p 19. (32) Ramirez, R.; Bohm,M. C.Int. J . Quantum Chem. 1986. 30, 391.
A Charge-Iterative Hamiltonian for Molecular Electronic Spectra John David Baker and Michael C. Zerner* Quantum Theory Project, University of Florida, Gainesville, Florida 32611 (Received: June 19, 1990)
A complete charge-dependent INDO model Hamiltonian appropriate for the calculation of molecular spectra is developed and examined. A procedure for the acquisition of atomic and molecular parameters is presented that makes use of readily available experimental data. Simplified parameter functions, allowing more information to be incorporated into the model with less effort than is the case for previous parametrization strategies, are suggested. The charge-iterative model forwarded here surpasses the current spectroscopic INDO Hamiltonian in the quantitative prediction of n-r* excitation energies for carbonyls without loss of accuracy in the prediction of r--A*excitation energies.
I. Introduction Semiempirical Hamiltonians for electronic structure calculations owe their efficiencies to reducing the number of integrals processed to M , where N is the size of the basis set. In so doing, matrix multiplications and diagonalizations, both M steps, dominate, at least at the Hartree-Fock (HF) level. In addition, most semiempirical methods utilize a minimum basis set (MBS) of valence-type orbitals, further reducing N in the more complex N3 steps. These models compensate for the reduction of integrals and small basis sets by parametrizing directly on atomic phenomena and molecular properties, and are thus often able to reproduce experimental values in a more accurate fashion than the MBS a b initio calculations on which they are modeled.1-2 The MBS assumption is definitely a restriction, as it does not properly allow the orbitals associated with an atom in a molecule to expand or contract with gain or loss of electrons, and more or less freezes the potential of an atom in a molecule to that seen in neutral atoms. In this paper, we reexamine the atomic parameters that enter into most semiempirical Hamiltonians, borrowing from concepts that seem natural in density functional theory? Density functional theory suggests atomic parameters that are functions of charge
and atomic number. We present an initial attempt at utilizing such parameters in a fashion appropriate for the calculation of electronic excitations and ionizations. Due to the success of the spectroscopic version of the intermediate neglect of differential overlap (INDO/S) approximation in the prediction of electronic properties, we seek to extend the basic form of this Hamilt~nian.~ At the heart of many semiempirical models is the determination of purely atomic parameters from experimental atomic information. These atomic parameters, in turn, help define the atomic potentials that electrons in the molecule realize. We expand the total atomic energy (E) in a Taylor series with respect to the total number of electrons N E(N,Z) . . = aE aZE E ( Z , Z ) + ( N - Z ) - + Y2(N- Z)2- + higher terms (1) aN ann and utilize the concepts of hardnessS and electronegativity or chemical potential6*'
electronegativity = x
(1) Sadlej, J. Semi-Empirical Methods of Quantum Chemistry; Wiley:
New York, 1985. ( 2 ) Pople. J. A.; Beveridge, D. L. Approximate Molecular Orbital Theory; McGraw-Hill: New York, 1970. (3) Parr, R. G.; Yang, W . Density-Functional Theory of Atoms and Molecules; Oxford: New York, 1989.
aE
= --d N
= -chemical potential (3)
Ridley, J.; Zerner, M . Theor. Chim. Acta 1973, 32, 111-134. (5) Peanon, R. G. J . Chem. Ed. 1987,64, 561-567. (6) Pritchard, H. 0.; Skinner, H. A. Chem. Reu. 1955, 55, 745-786. (4)
0022-3654/91/2095-2307$02.50/00 1991 American Chemical Society
Baker and Zerner
2308 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 and derive, in terms of atomic charge Q = Z - N E(Q,Z) = E(0,Z) + Qx
+ Q2q + ...
(4)
On the other hand, the energy for the average configuration of an atom containing only s and p electrons in the valence shell is given by8 E(N,,Np,Z) N,V,
+ NpVp+ !lz(Ns + Np)(Ns+ Np- 1)F” !16NsNpG1- !125Np(Np-
1)P (5)
in which Usand Upare the one-electron energies for s and p electrons9and F”, GI, and P are the Condon-Shortley two-electron integrals.aJO In deriving eq 5, we have assumed that F’= ~ ( s s ) = P(SP) = P(PP). Parameters in the INDO HF model represent integrals. Once these integrals are chosen via computation in an appropriate atomic basis, or in this case by parametrization, the solution of the molecular problem proceeds along well-established lines whether the theory is H F or correlated. In a normal HF procedure, the charge density is varied until self-consistency is obtained. The integrals are fixed, and therefore it is through fixed integrals that the H F procedure “sees” the potential created by the centers in the molecule. In large basis set calculations, linear combinations of these integrals over the basis allow the potential to vary with molecular environment. This degree of flexibility is never present in any minimal basis set calculation, and it is this deficiency that we would like to correct. In a recent paper, we reviewed a method for obtaining total energies from experimental data and described an atomic parametrization that is fully charge dependent and is capable of modeling atomic energies very accurately, although it remains in a MBS formalism.” As a result of predicting energies accurately, orbital electronegativities and hardness values are also reproduced. We denote this approach “energy partitioning” since the total atomic energy is modeled. This procedure is most appropriate for the calculation of energy-based properties such as molecular geometries. Energy partitioning, however, requires that derivatives of the parameters be taken with respect to Q as part of the total energy derivatives in order to reproduce atomic potentials. The H F procedure within a MBS restriction does not include this effect. In the present treatment, a slightly different approach is pursued. Once the electronic occupation of an atom in a molecule is defined, the atomic parameters for that center are fixed such that the slope and curvature of the atomic energy of interest are maintained with frozen parameters. If the atomic occupations change, then the atomic parameters of interest are adjusted and again frozen to reflect the new slope and curvature of the energy at that occupation. We denote parameters chosen in this way “density partitioning”, since parameters are adjusted to reflect the instantaneous chemical potential and hardness at a given occupation or density of a center. Density partitioning of atomic energies allows the HF procedure to =seen the bare-atom potentials directly. The idea of introducing charge-dependent parameters into approximate electronic structure theories is, of course, not a new onell-17 and dates back to the work of Wheland and Mann,12who (7) Parr. R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J . Chem. Phys. 1978, 68, 3801-3807. (8) Slater, J . C. Quantum Theory o j Atomic Srructure; McGraw-Hill: New York. 1960. (9) Karlsson, G.; Zerner, M. fnr. J . Quuntum Chem. 1973, 7, 35-49. (10) Hinze, J.; Jam, H. H. J . Chem. Phys. 1963,38, 1834-1847. ( 1 1 ) Baker, J. D.; Zerner, M. C. J . Phys. Chem. 1990, 94, 2866-2872. (1 2) Wheland, G. W.; Mann, D. E. J . Chem. Phys. 1949, 17, 264-268. (13) Brown, R. D.; Heffernan, M. L. Truns. Furuduy SOC. 1958, 54, 757-764. (14) Parks, J . M.; Parr, R. G. J . Chem. Phys. 1960, 32, 1657-1681. (15) Iffert, R.; Jug, K. Theor. Chim. Actu 1987, 72, 373-378. (16) Kovesdi, I. THEOCHEM 1987, 152, 341-346. (17) Singerman, J. A.; Jaffe, H. H. J . Am. Chem. Soc. 1981, 103, 1358-1360.
convincingly demonstrated its utility. In work with motivation related to ours, Singerman and Jaffe” recognized that it was difficult to reproduce the spectra of sulfur-containing compounds by using the CNDO/S methodla without creating charge-dependent parameters for the sulfur d orbitals. Our approach differs from others in that we attempt to generalize the concept of charge dependency for the whole Hamiltonian in a fashion consistent with a more accurate description of the local potentials of all the atoms in a molecule and not just to correct for specific cases for which charge dependency is a necessity. In section 11, the method for obtaining chargedependent atomic parameters is discussed. The concept of charge dependence is extended to the resonance integrals and two-center Coulomb integrals of the INDO molecular Hamiltonian4J9 in section 111. In section IV, some comparisons are made between the INDO/S approximation and the charge-iterative approach presented here. The Condon-Shortley integrals GI and P are small relative to the other atomic parameters. The formulas for these integrals are taken from our previous work.” 11. Atomic Hamiltonian
Consider the energy expansion truncated at second order: E(N,Z) = A(Z)
+ NB(Z) + MC(Z)
(6)
In terms of this expression, x and q can be written as
x = -B(Z) - 2NC(Z) q = 2C(Z) (7) We can take derivatives of eq 5 with respect to the number of p electrons and obtain
1% J c
-xp =
.
.
=
and r
(9)
One can now solve for the spherically symmetric Coulomb integral F’ in eq 9 and note that from finite differences q = 1/2(IP- EA). r
(10) The P integral is small with respect to the leading term of the ionization potential (IP) less the electron affinity (EA). Ignoring the small P term, we recover the well-known Pariser approximationZ0for P. The finite difference approximation for xp is 1/2(IP EA). Solving for V, in eq 8 yields
+
Up = -xp - v2(2N,
+ 2Np - 1)P + l/gG1 4- y25(2Np - 1)P (1 1)
Using finite difference approximations for xpand F’,one obtains well-known and in this case formally equivalent expressions for Upand similar expressions for Us.9*19
up= -IP - (Ns+ Np- 1)P+ ‘/6N,G1 + 2/25(Np- 1)p = -EA - ( N , + N p ) P+ t/dV,Cl
+ Z/5P
(12)
From the above discussion, it is clear that if the IP and EA of an atomic orbital are known, then the parameters associated with (18) (a) Del Bene, J.; Jaffe, H. H. J . Chem. Phys. 1968.48, 1807-1813. (b) Ellis, R. L.; Kuehnlenz, G.; Jaffe, H. H. Theor. Chim. Actu 1972. 26,
131-140. (19) Pople, J. A.; Beveridge, D. L.; Dobosh, P. A. J . Chem. Phys. 1967, 47, 2026-2033. (20) Pariser, R. J . J . Chem. Phys. 1953, 21, 568-569.
The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 2309
A Charge-Iterative Hamiltonian for Electronic Spectra
that orbital can be uniquely determined. Alternatively, if we know the instantaneous derivatives of the energy with respect to charge, the same analysis can be performed. Implicit in the above derivatives is the assumption of continuously differentiable charge. With this assumption, the experimental energies can be fitted to a least-squares polynomial and the derivatives taken explicitly. This is the approach used for the determination of F’. There are several benefits to this approach. First, only total atomic energies are necessary instead of both I P S and EA’s. Electron affinities are not as available as total energies. Second, this approach allows the fitting of a single polynomial expression for the atomic energies as a function of atomic number and charge for a whole row. With a single expression, derivatives can be taken for a given atom and charge for which specific atomic information may not be available. Finally, the P s obtained in this way are of the same form as those obtained from Slater type orbitals. One-center FO‘s with a single exponent using Slater s type functions can be exactly expressed as a constant dependent on the principal quantum number of the shell multiplied by the orbital exponent. Using Slater’s rules2’ for orbital exponents would give an expression F‘ = KJ = K,(Z - a)/n = A BZ + CQ. The principal quantum number is denoted by n, and u is the atomic screening constant. In units of electronvolts K , = 17.0062, K2 = 9.8856, and K3 = 7.0244. For n = 2 and 3, 35 and 32 data points, respectively, were used to fit the energy polynomial. The second derivatives were taken and used to fit F’ according to eq 10 to obtain
+
n =2
F’ (eV) = 3.04282 + 1.809722, + 3.28219Q
n=3
F’ (eV) = 2.70856 + 1.113492, + 1.22155Q
(13)
Z, is the valence charge (e.g. C = 4), and Q is the net charge. The standard deviation for both models is 0.1 eV. This form of F’ appeals to our chemical intuition. As charge increases, we expect the center to contract and thus repulsions to increase, as predicted here. The slope of a plot of this variation with charge is smaller for second-row elements, as we would expect for larger more polarizable atoms. If we divide by K,,, an effective exponent for an atom can be obtained that is of the same form as that obtained from Slater’s rules, and thus we can extract screening constants. The so-called core terms, Usand Up,are the one-electron energies consisting of kinetic energy, nuclear electronic attraction, and a core-valence potential due to the neglect of core electrons.22 By use of single-exponent Slater orbitals, the kinetic energy and nuclear attraction portion of the core terms can be calculated exactly as U = A’(n,l) t2- B’(n) tZv+ ( V,). A’is a constant dependent upon the principal and angular momentum quantum numbers, B’is a constant dependent upon the principal quantum number only, and (V,) is the core-valence potential energy. Expanding the exponent, as with F’,one obtains a general form for the core terms.
+ b2, + cZV2+ ( d + e2,)Q + fQz + (V,) = u ’ + bZ, + cZ: + ( d ’ + e’Zv)Q +f’Q2
Ui = a
(14)
The second equality assumes that the core-valence potential does not in itself generate any higher terms in 2,or Q. Since atomic ionizations are of interest, we have chosen to use eq 12 to obtain values for the core terms. Only IPSare necessary to perform the analysis, as F‘has already been determined. Fitting in this way reproduces the finite difference approximation for electronegativity, which is equivalent to Mulliken’s scale,23and also reproduces Koopmans’ approximation for IP‘s and EA’s for atoms. Again, data for a whole row are collected to fit eq 14 so that interpolation for missing data allows core terms to be calculated. The modeled core terms in electronvolts are (21) Slater, J. C. Phys. Rev. 1930, 36, 57-64. (22) Zerner. M . Mol. Phys. 1972, 23, 963-978. (23) Mulliken. R. S.J . Chem. Phys. 1934, 2. 782-793.
+
U2s = 1.747 - 5.0572, - 1.916ZV2+ (2.609 - 3.619Zv)Q 1.967@ no. of pts = 30 std dev = 0.21 eV
U2, = 0.996 - 2.7412, - 1.8762,’
+ (1.906 - 3.3172,)Q + 1.635@
no. of pts = 35
std dev = 0.18 eV
Ujs = 0.296 - 4.2672, - 1.1892;
(15)
+ (-0.186 - 1.2462,)Q + 0.697Q2
no. of pts = 23 U3, = 0.761 - 2.3562, 1.2422;
no. of pts = 36
std dev = 0.30 eV
+ (0.152 - 1.080ZV)Q+ 0.565Q2 std dev = 0.40 eV
(16) It is appealing that the model suggests only a quadratic term in 2,for a fixed charge. Under current methods of parametrization, each atom in a row has to be parametrized separately for each charge. For a given s,p shell in a row at a fixed charge, one would require 16 separate parameters. In this model anions, neutrals, and several levels of positive charge are modeled with just 12 parameters. For hydrogen, we only have two useful energy values (neutral and anion) and two unknowns, namely F’ and Us.We set F’ = IPS - EAs = 12.841 eV and Us= -IPS = -13.585 eV. 111. Molecular Hamiltonian
There are two types of terms that must be addressed in a parametrization of two-center quantities a t the INDO level of approximation. These are the two-center repulsion integrals and the one-electron bond parameters or resonance integrals. The generalized form of many two-center repulsion integrals used in semiempirical theory can be expressed as
In this equation, f’ is known as the Weiss factor4 and RABis the internuclear separation. YAB(O) is the limit of for R A B = 0. YAB(O) is usually expressed as an average of the two one-center Ps.For this study, the one-center limit is calculated exactly by using Slater orbitals with effect exponents obtained from F’ in section 11. The Weiss factor is set equal to 1.2 and m is fixed at 1, according to the prescription of Mataga and N i s h i m o t ~ . ~ ~ These values were fixed in this study, but in future parameter studies, both fg and m will be allowed to vary in the parametrization. The INDO two-center one-electron matrix element denoted is usually expressed as the average of one-center bond parameters weighted by the overlap (Sij)between two Slater orbitals and an appropriate fixed interaction factor Uj) for the type of orbitals invol~ed.~*~~
The p p interactions are split between the contributions from pure u interaction and pure P interaction. The Euler rotation factors are denoted by g, and g,. Parametrization of this matrix element involves determining the one-center bond terms and appropriate interaction factors. Encouraged by the simple form of the core terms in section 11, we seek to parametrize the atom bond variables with the same (24) Mataga, N.; Nishimoto, K. Z . Phys. Chem. (Munich) 1957, 13, 140-157. (25) Del Bene, J.; Jam, H. H. J . Chem. Phys. 1968, 18, 1807-1813.
2310 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991
Baker and Zerner
form as that of eq 14. Standard practice for the determination of resonance integrals requires separate parametrizations for each atom. This requires reliable molecular information for molecules of each atom in a row. In this study, the polynomial for a row is parametrized at once. Molecular ionization potentials and excitation spectra for several compounds were used in order to adjust the bond parameter polynomials and interaction factors. The results are presented in section V and compared to the current INDO/S Hamiltonian. For the present, only hydrogen through fluorine have been examined. The term quadratic in charge for principal quantum number 2 was not included due to the relatively few compounds studied.
BH (eV) = -6.527 - 10.157Q + 5.129Q2 6Li-F
TABLE I: Electronic Excitation Energies (em-') for tbe Model Compounds charge molecule excitation exptl INDO/S iterative benzene' vu* 38090 37797 38060 benzene vu* 48972 48370 48483 benzene vu* 55900 54445 54757 benzene u-u* 55900 54445 54757 pyridine4 n-r* 34771 35500 34337 pyridine A-u* 38350 38686 38465 pyridine n-u* 43911 45294 pyridine 7-77 * 49750 49430 49807 pyridine u-u* -55000 55761 55819 pyridine u-u* -55 000 56 432 56 510 crotonaldehydez8 n-r* 30490 24729 28264 crotonaldehyde T-u* 45454 47182 45638 formaldehydez9 n-r* 33898 25653 30370
(19)
(eV) = 2.351 - 2.6612, - O.6716Zv2 + (0.503- 5.0662,)Q (20)
f,, = 0.8976
f,, = 1.166
f,, = 0.5283
(21)
TABLE 11: Koopmans' IP's (eV) for the Model Compounds INDO/S charge-iterative molecule exptl (Koop) (KooP) 9.25 ( u ) benzene30 8.95 9.39 benzene 11.49 (a) 12.47 12.60 12.1 ( u ) 13.32 13.76 benzene 15.25 14.70 13.8 ( u ) benzene 11.36 (u) BNH," 11.35 1 1.47 BNH4 12.08 (a) 12.69 12.52 14.12 ( u ) BNH, 15.63 15.38 10.51 (u) 9.99 10.38 CZH,)O
IV. SCF/CI A modified self-consistent field (SCF) program26 has been developed to incorporate charge iteration. The algorithm is as follows: (1) Assume all atoms in a molecule are neutral. (2) Evaluate all parameters and necessary integrals for performing the S C F calculation. (3a) For the first SCF sequence, cycle until density is adequately converged to obtain a population analysis. (3b) For the final S C F sequence, stop at normal convergence on energy, density, or both: then exit the S C F routine. (4) Assign charges to centers via population analysis and compare with the previous distributions. If they are the same to within a given tolerance, stop; if not, go to step 2. Normally only two S C F sequences are necessary to observe stable atomic charges. The diagonal elements of the Fock matrix include a term that can be evaluated as
for a basis function on center A. QB is the effective charge at center B seen by center A. Since the PBBterms represent the LOwdin2' populations for B centers, these terms are used to define atomic charge. Exponents for the overlap portion of the resonance integrals are those of Slater with the exponent for hydrogen set at 1.2. Due to the fact that standard INDO/S calculations usually do a poorer job in estimating n d * excitation energies relative to 7 ~ energies, a modification to the way the core terms are evaluated has been developed to overcome this problem. Consider the noneffectual modification to eq 5: E(Ns,Np,Z)= Ns(Vs+ A )
( 2.1.
+ Np
Up -
Y2(Ns + Np)(Ns+ Np- 1 ) P - !/"NsNpG'I/z5Np(Np- 1 ) P = NJJ,' + N,Up' +
... ( 2 3 )
These modified core terms do not affect atomic properties, but they do have an effect on the hybridization of a center in a molecular calculation. By making A positive, one moves the (26) Szabo,A.; Ostlund, N. S.Modern Quantum Chemistry; Macmillan: New York, 1982. (27) Ldwdin, P . 0. J . Chem. Phys. 1950, 18, 365-375. (28) Bayliss, N. S.;McRae, E. G. J . Phys. Chem. 1954,58, 1006-1011. (29) Suzuki, H. Electronic Absorption Spectra and Geometry of Organic Molecules; Academic: New York, 1967. (30) Rabalais, J. W. Principles of Ultrauiolet Photoelectron Spectroscopy; Wiley: New York, 1977. (31) Westwood, N. P. C.; Werstiuk, N. H. J . Am. Chem. Soc. 1986,108,
891-894. (32) Dekock, R. L.Electron Spectroscopy: Theory, Techniques and Applications; Brundle, C . R., Baker, A. D., Eds.; Academic: New York, 1977; Vol. 1. (33) o h m , Y.; Born, G. Adu. Quantum Chem. 1981, 13, 1-88. (34) Brundlc, C. R.; Robin, M. B.; Kuebler, N. A. J . Am. Chem. SOC. 1972, 94, 1466-1475.
C2H4 C2H4
F p
FZ FZ
formaldehyde" formaldehyde formaldehyde formaldehyde H p ~yridine'~ pyridine pyridine pyridine pyridine
Li22 C22 *
12.85 ( u ) 14.66 ( a ) 15.83 (u) 18.8 ( u ) 21.0 (a) 10.9 (n) 14.5 ( u ) 16.2 ( u ) 17.0 ( u ) 15.88 ( u ) 9.67 (n) 9.8 ( u ) 10.5 (u) 12.45 ( u ) 13.1 (u) 4.96 ( u ) 12.0 ( u )
13.52 16.09 16.91 19.2 23.51 10.82 13.44 16.42 19.2 18.87 10.09 9.06 9.89 13.01 14.06 5.1 1 11.08
13.06 15.73 17.20 19.5 1 23.23 12.22 13.99 14.74 20.29 15.73 10.49 9.54 10.04 13.02 14.27 4.80 11.37
one-electron s-orbital energies closer to the p-orbital energies, allowing a stronger mixing of the two. A pronounced effect is observed on the nonbonding orbital energies relative to those of -A bonding. The use of this scheme is only valid in a charge-iterative scheme, since the ratio N J N , is also modified during the calculation to maintain equivalence between eqs 5 and 23. The value of A for oxygen was set at 7 eV. This value was determined as a compromise between correcting for n--A* excitation energies and maintaining an accurate prediction of molecular ionization energies. To obtain the lowest lying excitation energies of several of the test compounds, a limited singles-only configuration interaction (CI) was performed. The CI calculations were limited to seven active occupied and seven active virtual orbitals. In addition, we report I P S estimated according to Koopmans' theorem. The INDO/S model does a good job of reproducing ionization energies when a CI is performed on an ion, and these results are already reflected at the Koopmans level. V. Results The results for the model compounds used for electronic excitation spectra are presented in Table I and compared to the INDO/S approximation. The root-mean-square error for the INDO/S model is 3194 cm-I, and that for the charge-iterative model is 1460 cm-l. Most noted is the substantial improvement in n-u* carbonyl excitation energies for the charge-iterative model. The results for the model compound I P S are presented in Table 11, in which the root-mean-square error for the INDO/S and the
J. Phys. Chem. 1991,952311-2316 charge-iterative models is 1.18 eV. No quantitative improvement is observed for the Koopmans IP’s of the model compounds for the two parameterizations. Both models maintain correct qualitative ordering for the orbital energies except for pyridine. In this case, both models place the nonbonding orbital lower in energy than is observed relative to the occupied ?r orbitals. This order is corrected in AE(SCF) calculations or through a CI treatment of the positive ion. We are currently examining the use of electron propagator35methods to correct for such disarrangement by including low-order correlation and orbital relaxation effects associated with ionization. VI. Conclusions This study has demonstrated that improvement over existing model spectroscopic Hamiltonians is possible with charge-dependent parameters. While the quality of r-r* transitions is maintained, the simple and physically intuitive strategy for improving n-?r* excitation energies for carbonyls is possible. In addition to improving agreement with molecular electronic spectra, the information content of atomic parameters is increased while their structures are simplified. By fitting for a whole row a t once for all available charges, both the one-center Coulomb integrals and one-electron core integrals are easily obtained with (35) Linderberg, J.; Ohm, N. Y. Propagators in Quanrum Chemistry; Academic: New York, 1973.
2311
total atomic energies or total energies and ionization potentials. It is not necessary to obtain orbital electron affinities as is necessary for the use of Pariser’s approximation for Coulomb integrals. Since the parameters are constrained to be at most quadratic in atomic number, interpolation for sparse atomic data occurs naturally, as does extrapolation to atomic anions. The extension of the form of the one-center core terms to the one-center resonance terms greatly simplifies the parametrization at the molecular level. Much smaller data bases of molecular data can be employed to obtain parameters. If no charge dependence for the resonance integrals is needed, one can define the resonance integrals for a whole row by specifying only three values (preferably for the atoms at the ends of the row and one in the middle). The analysis of the atomic parameters in terms of the chemically useful concepts of orbital hardness and electronegativity provides a connection to the areas of density functional theory.36 This type of analysis allows the probing of forces within molecules necessary for the description of chemical activity and in this case orbital energies and electronic spectra. Acknowledgment. J.D.B. acknowledges support through a Tennessee Eastman Postgraduate Fellowship. This work was supported in part through a grant from Eastman Kodak Co. and from the Office of Naval Research. (36) Gizquez, J. L.; Ortiz, E. J. Chem. Phys. 1984,81, 2741-2748.
Electrostatic Energy In Chloride Salts of Mononltrogen Organic Bases Jacek Lubkowski and Jerzy Blaiejowski* Institute of Chemistry, University of Gdatisk, 80-952 GdaAsk, Poland (Received: June 26, 1990)
The electrostatic part of the lattice energy in chloride salts of mononitrogen organic bases was determined by adopting the Ewald method. The calculations were performed for compounds for which complete, or at least partial, crystal structures are known. In the case of incomplete structures, the MNDO geometry optimization procedure was used to find the unknown positions of atoms. The electrostatic energy calculations were performed on the assumption that a negative (-1) charge is attached to the chlorine atom and a positive (+I) charge either is located on the N atom or is distributed among all the atoms in the cation. The charge distribution in the isolated cation was evaluated by applying CNDO/2, INDO, and MNDO methods. The electrostatic energy in an NH4Cl crystal was also evaluated by considering the influence of neighboring ions on the charge distribution in the lattice. The electrostatic energy values derived were compared with the published values of the crystal lattice energy determined either theoretically or experimentally. The agreement appeared to be satisfactory, which implies that, in the case of the compounds studied, the main contribution to the cohesive forces is made by the Coulombic interactions.
ions; E,, the repulsive interactions; Ed, the van der Waals inter-
actions; and Eo, the zero-point energy. Knowledge of the charge distribution in the lattice enables a precise evaluation to be made of the E,, term on the basis of the Coulomb law. For lattices composed on K,A, units, one can assume that point charges of values an+ and am-, for cation and anion, respectively, are located at the centers of given atom^.^^^ For simple ionic substances containing symmetrical ions, for example octahedral ions, the charge distribution between atoms can sometimes be guessed.2.6 In the case of compounds composed of unsymmetrical ions, the charge distribution is not easily predictable. For the latter substances, therefore, theoretical calculations of the lattice energy have not so far been carried out. To this group belong chloride salts of mononitrogen organic bases. A somewhat more difficult problem is posed by the evaluation of the remaining three terms on the right-hand side of eq 2. The
(1) Thakur. K. P. Ausr. J . Phys. 1976. 29, 39. (2) Jenkins, H. D. B.; Pratt, K. F. Adu. Inorg. Chem. Radiochem. 1979,
(4) Weenk, J. W.; Harwig, H. A. J. Phys. Chem. Solids 1977, 38, 1047. ( 5 ) Atkins, P. W. Physical Chemistry, 3rd ed.;Freeman: New York, 1986;
Introduction Crystal lattice energy determines the magnitude of cohesive forces in the solid phase. In the case of an ionic substance of general formula K,,,A,, the crystal lattice energy (E,) is defined as an energy change for the process where a is the multiplicator accounting for the actual valences of both ions. It is generally recognized that four effects contribute to the crystal lattice energy.’” These effects are expressed by eq 2, where Eelrepresents the electrostatic interactions between E, = -Eel + E, - E d + Eo (2)
22, 1. ( 3 ) Raghurama, G.;Narayan, R. J . Phys. Chem. Solids 1983, 44, 633.
0022-3654/91/2095-2311$02.50/0
p 595. (6) Jenkins, H. D. B.; Pratt, K, F. Prog. Solid Stare Chem. 1979,12, 125.
0 1991 American Chemical Society
