2111
11/10THEORY OF EDA COMPLEXES
Molecular Orbital Theory of Electron DonorAcceptor Complexes. 11. Charged Donors and Acceptors by R. L. Flurry, Jr. Department of Chemistry, Louisiana State University in New Orleans, New Orleans, Louisiana (Received March 14, fg88)
70188
A consideration of the linear combination of molecular orbitals (LCMO) treatment of electron donor-acceptor (EDA) complexes’ as applied to charged donors and acceptors is made. The explicit form of the Hamiltonian operator for a one-electron description is considered in order to define the electrostatic interaction term. It is seen that the nature of this term differs for differing charge types of donor and acceptor. Calculations for the neutral donor, neutral acceptor, the neutral donor, positive acceptor, and the negative donor,,positive acceptor types are presented. In all cases the agreement with the available experimental data is satisfactory. In the Appendix a formal justification of the one-electron equationsis given in terms of a polyelectron molecular orbital treatment and in terms of a polyelectron valence bond treatment.
I. Introduction I n paper I of this series’ (hereafter referred to as I), the linear combination of molecular orbits (LCMO) description of electron donor-acceptor (EDA) complexes was presented. The interaction of the donor and acceptor was assumed, as a first approximation, to arise primarily from the interaction of the highest occupied molecular orbital of the donor (HOn!tO)D with the lowest empty molecular orbital of the acceptor (LEMO)A. The resulting complex orbital $DA was assumed to be a linear combination of these two molecular orbitals $DA
=
U$D
+ b$il
(1)
where $D is the (HOMO)D and $A is the (LEMO)Aand a and b are mixing parameters to be determined by the variational principle. It was shown that within this approximation, and with the further approximation of zero differential overlap (ZDO), the ground-state energy and the energy of the first charge-transfer transition AECTcould be expressed as
- b2EAa + ~ U ~ P D Ab2V,s (az - b 2 ) ( I ~ - E A A - V e s ) - 4abPDA2
EN = AECT =
-a21D
(2)
(3) where ID is the ionization potential of the donor, E’AAis the electron affinity of the acceptor, &A is the “resonance” interaction between the donor and acceptor, and - Ve8an electrostatic interaction term, defined such that it represents the interaction which would arise if a unit electron charge were transferred. The present paper is directed a t a more detailed study of this V,, term. 11. Quantum Mechanical Interpretation of the Electrostatic Interaction Parameter The simplest approach for gaining insight into the
nature of the electrostatic interaction parameter (Vaein I) is to consider explicitly a one-electron Hamiltonian operator (H) for the complex and for the isolated donor and acceptor. The simplest analogy for such a system would be the one-electron MO treatment of a diatomic molecule. Consider two nuclei, which we shall call D and A. The Hamiltonian for one electron located on D would be (in atomic units) HD
=
-1/2V2
-rD
(4)
where V2 is the Laplacian operator describing the kinetic energy of the electron, Z D is~ the ~ effective nuclear charge of D, and ?“D is the distance from the electron to the nucleus. The Hamiltonian for one electron on the acceptor would be completely analogous. H A
=
--,/2V2
p - ZrA
(5)
The one-electron Hamiltonian for the united complex would be
where RDAis the separation of D and A. It must be recognized that in electron donor-acceptor complexes the electron would be associated only with the donor if there were no interaction, and that the magnitudes of the interactions are usually relatively small. Even in the complex, the electron will spend most of its time on the donor; consequently, the complex should be fairly well described by the wave function of the donor. For this (1)
R.L.Flurry, Jr., J.Phys. Chern., 69, 1927 (1966). Volume 79, Number 7 July 1089
R. L. FLURRY, JR.
2112 reason, we could write the Hamiltonian of the complex as the sum of H D and a perturbation.
HDAe= H D
+v
(7)
where
The use of a Hamiltonian of the form of eq 4 with only the donor wave function would lead to the conventional perturbation theory treatment of EDA I n this work, however, the ground-state wave function for the complex orbital arising from (HOMO)D and (LEMO)Ais expressed as a linear combination of the donor and acceptor wave functions as in eq 1. The energy of the ground state of the complex will then be (using Dirac notation and formally neglecting overlap)
EN =
(U+D
+ ~+AIHDAIu+D+ b + ~ )
(9)
(9%) The first two integrals to the right of the equals mark in eq 9a are the expectation values for the one-electron energies of D and A, respectively. The third integral is the PDA of I. The fourth and fifth integrals represent the attractive force of A on an electron in the orbital described by +D and the attractive force of D on an electron in the orbital described by +A, respectively. The final term is just the internuclear repulsion term. These three last terms make up the parameter - V,, in paper I, If, now, it is recognized that in the electron donor-acceptor systems normally considered the donor and acceptor are not bare monatomic nuclei, but are polyelectron, polynuclear species, usually in their singlet ground states, eq 9a, with some modifications, can give an insight into their description. It is generally accepted that the orbitals contributing most greatly to EDA interaction are usually the highest occupied orbital of the donor and the lowest unoccupied orbital of the acceptor. I n the above treatment, then, these would be represented by +D and +A. If a Hartree-Fock (HF) solution could be obtained, the eigenvalue corresponding to the highest occupied orbital obtained by the H F donor Hamiltonian operating on #D can be approximated by the negative of the ionization potential of the donor.4 (+AlHA[+A) should similarly approximate the negative of the electron affinity of the acceptor. These are analogous to the case of the bare The Journal of Physical Chemistry
nuclei. The additional terms in (sa), however, require some additional interpretation. If D and A are neutral entities having closed-shell /RDA be relaground states, the term Z D ~ ~ Z A ~ ~should tively small. Such effects as van der Waals forces, dipole interactions, dipole-quadrupole interactioq6etc., could be included in this term. Since A will bear no net charge when the electron being described by eq 9a is on D, the term a2ZAeff(+Dll/?Aj+D) should essentially vanish. The remaining term, b 2 Z ~ e f f ( $ ~ l l / r ~isl # ~ ) , thus the main contributor to the lie,of paper I.6 This term can have appreciable magnitude since it represents the interaction of the electron in the vicinity of A with D which will now bear a charge. For systems in which D or A, or both, are charged species, the contributions to V,, can be obtained by similar reasoning. The “effective nuclear charge” term must be visualized as the net electrostatic interaction of the donor and acceptor with one electron removed from the highest occupied orbital of the donor. It will be of significant magnitude when the result of this process leaves both the donor and the acceptor with a net charge. This occurs only when we are dealing with a complex between a neutral donor and a positively charge acceptor. Table I presents these contributions for t,he most reasonable cases. Table I : Contribution of Various Terms to V,, for Various Charge Types of Donor and Acceptor“
Ob -0 0
-
0 0
Small Small
Large Small
+
Large Large
Large Small
Ha,
Hbb
Contributes to
Small
Small Large Small Total energy
The terms indicated as “small” may, to a first approximation, be neglected when constructing the appropriate elements of the secular determinant, * No net charge. Negative charge, Positive charge.
Application of the variational principle t o eq 9 yields eq 10 as the secular determinant to be solved. The
(2) J. N. Murrell, J . Amer. Chem. SOC.,81,5037 (1959); Quart. Rev.
(London), 191 (1961). (3) M.J. S.Dewar and A. R. Lepley, J . Amer. Chem. Soc., 83,4560 (1961). (4) T.A. Koopmans, Physica, 1, 104 (1933). (5) M. W.Hanna, J . Amer. Chem. Soc., 90,285 (1968). (6) Note that the (admittedly deficient) basic assumption in this argument is that the “mobile” electron, when on either D or A, is effectively shielded by the localired electrons from any charge from the other center other than the net charge.
2113
A 4 0 THEORY OF EDA COMPLEXES
lowest root of this deteminant will be the ground-state energy of the complex in this approximation while the difference between the roots will be the energy of the charge-transfer transition. The Ha,, H b b , and Hab are the terms in eq 9a having coefficients a2,b2, and ab, respectively. consultation of Table I allows certain of these terms t o be neglected for the various charge types. The one-electron treatment as presented here can be shown to be a formally correct solution to the quantum mechanical description of donor-acceptor complexes if only the highest occupied orbital or the donor and lowest unoccupied orbital of the acceptor are considered. The proof of this is given in the Appendix, where interactions arising from orbitals other than ( H O i l l 0 ) ~and (LEMO)Aare also considered.
111. Method of Calculation The spectra and relative stabilities can be obtained from the root of the secular determinant shown in eq 10. The first charge-transfer transition energy is the difference between the two roots. The first approximation to the stabilization energy (AEstab) is the difference between the lowest energy root and the element Haa, The relative stabilities of a series can be obtained from the approximate expression of eq 11
where the superscript s refers to the complex chosen as the reference for the series. The method of choice of the PDA parameter has been modified somewhat from that used in I. I n I, the same value of PDA was used for all members of a given series. Obviously, as the complexes are varied this should change due to the change of the overlap between the donor and the acceptor. Since, however, the relative orientations of the donor and acceptor are not known, there is no meaningful way t o estimate the magnitude of this overlap. I t was decided, therefore, to weight the magnitude of PDA within each series by the amount of charge transferred in the ground state of each complex. The reasoning behind this choice for weighting PDAwith charge transfer was based upon the idea that the amount of charge transferred in the ground state of the complex and the spacing between the donor and acceptor should both be related to the degree of their interaction in the complex. The P DA term should be strongly dependent upon the spacing between the donor and acceptor, and hence, indirectly upon the amount of charge transferred. The method employed was t o assume an average PDAfor the series and t o divide this by the quantity (a2 - b 2 )for each member of the series. The average PDA was obtained as in I by a plot of AECT vs. the ionization potential of the donor. The slope of this plot yields (a2 - b2)average; the intercept is a function of the average PDA, dependent upon which charge type is under con-
sideration. The value of PDA so determined is the value used directly in I. I n the present work, this was multiplied by the slope in order t o obtain the /3 parameter to use for the charge-weighted P. The amount of charge transferred is relatively independent of the magnitude of PDA; consequently, the individual values of a and b for calculating the individual charge-weighted p’s were obtained by solving the secular determinant with the fixed average p. This procedure could be repeated iteratively t o obtain an “optimum” PDA; however, the improvement would probably not be worth the extra calculations. Although this is in no means to be considered an optimum method of choosing a PDA, especially when the amount of charge transferred is large, it does offer a distinct improvement over the pervious calculations. Unless otherwise noted, ionization potential values were taken from the tabulation by Kiser.’
IV. Examples Several examples of calculations involving neutral donors and aoceptors were presented in I. One example is repeated here to reveal the effect of the new p choice and for completeness, I n Table I1 are shown the calculated and observed charge-transfer transitions and relative stabilities for the complexes of the methylbenzenes with tetracyanoethylene (TCNE). The V e s parameter used here was estimated from an average of the interplanar distances for TCYE complexes by Boeyens and Herbstein* and the calculated charge distributions for the highest occupied orbital of benzene and the lowest unoccupied orbital of TCXE, assuming the out-of-plane Cp axis of TCNE and the Ce axis of benzene to be coincident. The electron affinity of TCNE was estimated from SCMO calculations performed in this laboratory. (The value of 2.1 eV is within the range of 1.53 to 2.2 eV obtained from estimates based on charge-transfer ~ p e c t r a . ~ )Note that the agreement between the calculated and experimental solution phase values for spectra and relative stabilities is satisfactory. The agreement with the recent gasphase valuesl0 is even somewhat better for the spectra. The tropylium ion complexes of Feldman and Winstein were chosen for the neutral donor, positive acceptor (0, +) type species.” The terms ZA“(+D/~/TA~+D) and ZA~~(+A~~/TD/$A) and Z A ~ ~ ( + D ~ / T A and + DZ) A ~ ~ (J/A~/?“D$A) which appear in Ha, and ”,, respectively (see eq 9a and Table I), should be of approximately the same magnitude, and of the same magnitude as the (7) R. W. Kiser, “Tables of Ionization Potentials,” U. 5. A.E.C. Publication TID6142, 1960. (8) J. C. A. Boeyens and F. H. Herbstein, J . Phys. Chem., 69, 2153 2160 (1965). (9) G. Briegleb, Angew. Chem. Int. Ed. Enol., 3, 617 (1964). (10) M. Kroll and M. L. Ginter, J. Phys. Chem., 69, 3671 (1963); M. Kroll, J . Amer. Chem. SOC.,90, 1097 (1968). (11) M. Feldman andS. Winstein, ibid., 83, 3338 (1961). Volume 73, Number 7
July 1069
R. L. FLURRY, JR.
2114
Table I1 : Spectra and Relative Stabilities of Complexes of the Methylbenzenes with Tetracyanoethylene, Charge Type (0,o)“
Benzene Toluene o-Xylene m-Xylene p-Xylene Hemimellitene Pseudocumene Mesitylene Durene Pentamethylbenzene Hexamethylbenzene
3.69 3.28 3.04 3.03 2.93 2.92 2.69 2.91 2.42 2.32 2.21
3.24 3.00 2.84 2.82 2.69 2.69 2.62 2.67 2.45 2.38 2.27
3.67 3.35 3.15 2.91 2.92 2.93 2.61 2.60 2.44
[OI 0.34 0.60 0.61 0.74 0.76 1.11 0.77 1.67 1.93 2.26
[OI 0.28 0.54 0.48 0.58
... [0.54]’
... 0.44
... 0.94 1.43 1.79 2.12
1.10 2.15
...
Empirical parameters: E A of TCNE, 2.1 eV (from SCMO calculations); ZD~~~($A/~/TD($A), 3.54 a Neutral donor and acceptor. eV; ( P D A ) ~-0.40 ~ ~ ~ eV. ~ ~ ~ A. , R. Lepley, J . Amer. Chem. SOC.,86,2545 (1964). Reference 10. R. E. Merrifield and W. D. Phillips, J . Amer. Chem. SOC.,80,2778 (1958). e Arbitrarily assigned reference value to put these numbers on the same scale as the two previous columns.
term (ZD”ZA~~)/RDA ; consequently, the electrostatic interaction terms should, to a first approximation, cancel out. This means that AECTshould be near, but somewhat larger than, the difference between the ionization potential of the donor and the electron affinity of the acceptor. This expectation is verified (Table 111).
series, then used with the entire series. The results are shown in Table 111. As would be expected, the spectral agreement for the polycyclics is poorer than for the benzene series. For the charge type (-,+), negatively charged donor and positively charged acceptor, the sodium and potassium halides were chosen. The spectra and dissociation energies of a number of these in the gas phase of are available.13a14 The integral ZA~~(+D~~/TA/+D) Table 111: Spectra and Relative Stabilities of Complexes of eq 9a may be evaluated by a point-charge approximaSome Aromatic Donors with the Tropylium Ion, tion, using the experimental equilibrium internuclear Charge Type ( O , + ) a distances. These are included in Table IV. A value of -0.51 eV is obtained for PDA for the sodium halides Log ----AEG, eV-(ID(K/KB), and -0.16 eV for the potassium halides. The electron Donor Calcd Obsdb E A A ) ~ calcd affinities of the alkali ions were taken to be numerically Benzene 4.05 4.07 3.24 (0) equal to the ionization potentials of the atornsl5 while Toluene 3.68 3.85 3.32 0.44 the ionization potentials of the halide ions were taken , . . 3.06 0.79 m-Xylene 3.45 to be numerically equal to the electron affinities of the p-X ylene 3.37 3.65 2.95 0.95 atoms. l 6 , I 7 The results are presented in Table IV. Mesitylene 3.32 3.40 2.90 1.05 The numerical agreement between the uncorrected Naphthalene 3.09 2.89 2.62 1.58 2-Methylnaphthalene 2.97 2.63 2.46 1.95 theoretical and the experimental results for the alkali Phenanthrene 3.03 2.92 2.53 1.78 halides is surprisingly good. This can be attributed Anthracene 2.69 2.36 2.05 3 24 primarily to two factors. First is the small amount of Pyrene 2.80 2.32 2.22 2.64 charge transfer occurring in the ground state of the a Natural donor, positive acceptor. Empirical parameters: molecules. Even in the vapor phase, the formulation Ed of tropylium, 5.5 eV (from SCRlO calculations), ( P D A ) ~ W ~ ~ ~ , M+X- seems fairly close to being the actual picture of -0.725 eV (from first five members of the series only). Referthe molecule. The second factor is the ability to calence 11. ‘ The ionization potential of the donor minus the culate reasonably well the magnitude of the electroelectron affinity of the acceptor.
Benzene and the four methylbenzenes reported (Table 111) should form a similar series for obtaining an average PDA value. The values of PDAfor the other donors reported should, however, not be expected to be similarly related due to the variations in size and shape of the molecules.12 Consequently, the average PDAwas determined as outlined in section I11 for the benzene The Journal of Physical Chemistry
(12) R.L. Flurry, Jr., J . Chem. Phys., 43,5203 (1965). (13) G. Herzberg, “Spectra of Diatomic Molecules,” 2nd ed, D. van Nostrand Co., Inc., New York, N. Y.,1950. (14) A. G. Gaydon, “Dissociation Energies,” Chapman and Hall, Ltd., London, 1953. (15) C. E. Moore, National Bureau of Standards Circular 447, Vol. 3, U.S.Government Printing Office, Washington, D. C., 1958. (16) T.L. Bailey, J . Chem. Phys., 28,792 (1958). (17) H.0.Pritchard, Chem. Rev., 52, 529 (1953).
nao THEORY OF EDA COMPLEXES
2115
Table IV : Spectra and Dissociation Energies of the Sodium and Potassium Halides, Charge Type ( -,-I-)" ---AECT,C
Compound I.,* eV
NaF NaCl NaBr NaI
KF KC1 KBr KI
3.56 3.75 3.51 3.24 3.56 3.75 3.51 3.24
Calod
5.48 4.50 3.96 3.26 4.88 4.59 4.09 3.40
evObsde
c - & , d
Calod
eV--l Obsdf
... ...
7.02 5.82 5.53 5.07 5.66 5.18 4.92 4.49
6.23 5.63 5.43 4.97 5.6 5.0 4.77 4.38
3.78 3.14 ,
..
4.47 3.92 3.32
ZA~"(*Dl/rA/$D),
eV
6.97 5.75 5.46 4.97 5.65 5.17 4.91 4.47
' Negatively charged donor and positively charged acceptor. Empirical parameters: E A for N a f , 5.14 eV; E A for K+, 4.34 eV (C. E. Moore, ref 15); PDAfor NaX, -0.51 eV; PDA for KX, -0.17 eV. * Ionization potential of the halide ion. References 16 and 17. Actually, the first electronic transition of the gasphase molecule. For dissociation into ionic species. Calculated as the dissociation energy into neutral species plus the ionization potential of the metal atom minus the electron affinity of the halogen atom. e Reference 13. Reference 14.
static interaction term involved. This is due to the fact of the spherical symmetry of the ions and the known internuclear separation in each case. The good numerical results in this idealized case lend credulence to the validity of the overall method. Experimental data for series of complexes of the (- ,0) charge types are very scarce; consequently, no series of this type is included. From an examination of Table I, however, it can be seen that the electrostatic interactions should not enter into the results. The value of AE~Tshould be greater than (ID- EA*). The stabilities should be smaller than for comparable systems of the (0,O) type or the (-,+) type. Briegleb and coworkers have studied a few complexes of this type.l8 Their results are consistent with this reasoning. For example, the bromide-TCNE complex in glycol dimethyl ether solvent has a AEcT value of 2.82 eV, while (ID - EA*) should be less than 2 eV. They found AHo for this complex to be essentially zero.
V.
Conclusions
tions were performed a t the Louisiana State University in New Orleans Computer Research Center (supported in part by N.S.F. Grant No. GP2964). Helpful discussions with Dr. P. Politzer are also acknowledged. Appendix. Polyelectron Treatment A. Molecular Orbital Formalism. Most calculations of the properties of EDA complexes concentrate their attention on the highest occupied orbital of the donor and the lowest empty orbital of the acceptor. Further, most of the common systems have closed-shell ground states. A better approximation for the description of EDA complexes would be a two-electron treatment, the two electrons under consideration coming from the highest occupied orbital of the donor. Designate the wave functions for the entire complex by the symbol x,for the donor by +,and for the acceptor by A. Lower case Greek symbols will be used for individual spatial orbitals while upper case characters will be used to indicate the total wave function. A bar will indicate a fi spin function and the absence of a bar an a spin function. The subscripts a,b. . . will be used for donor orbitals while i , j . . . . will be used for acceptor orbitals and u,v.. . . for orbitals from either the donor or acceptor. Roman numerals or upper case letter subscripts will be used to indicate an orbital of a particular electronic configuration of the entire system. For a numerical solution of polyelectron molecular orbital type problems it is convenient to work within the Hart,ree-Fock formalism. This permits the use of the one-electron Fock operator rather than the polyelectron Hamiltonian operator, and yields solutions in terms of one-electron orbitals. The Fock operator for a closed shell molecular system may be expressed 000
Fop(i)
=
Hcore(i)
Cb ( 2 J b ( i ) - Kb(i))
(AI)
where Hoore(i)consists of the (one-electron) kinetic energy and nuclear attraction operators and the J ( i ) and K ( i ) are the electron interaction operators, the coulomb and exchange operators, defined over molecular orbitals as Jb(i)$a(i)
=
(+b(j)
Il/Tjl$'a(i))#'a('$
(A2)
A further examination of the one-electron LCMO treatment of EDA complexes reveals that the electrostatic interaction term enters into the ground- and excited-state energies in different ways for different charge types. Numerical results are presented for the The agreecharge types (O,O), (O,+), and (-,+). ment with experiment is uniformly good, especially so for the sodium and potassium halides, the (-,+) charge type chosen.
The summation over b in eq A1 is over the occupied molecular orbitals. I n applying this to donor-acceptor complexes, the only difference would be in interpreting the integrals of eq A2 and A3 in terms of the orbitals of the entire complex. I n matrix notation, the Hartree-Fock equation for the complex will be
Acknowledgment. Financial support of this work came from the Petroleum Research Fund of the American Chemical Society (Grant No. 2796-A5) and from the Cancer Association of Greater New Orleans. Calcula-
(18) G.Briegleb, W.Liptay, and R. Fick, 2.Phys. Chem., 33, 181 (1962);2.Elektrochem. 66,851,859(1962). Volume 78, Number 7 July 1969
2116
R. L. FLURRY, JR.
where Fopis the Fock operator for the system, x is the matrix representation for the (one-electron) orbitals of the complex, and E is the (diagonal) matrix of one-electron energy eigenvalues. By analogy to the usual approximation methods for molecular orbitals, let us express the elements of the x matrix as a suitable linear combination of some starting set of basis functions $. This can most conveniently be done by a unitary transformation on the matrix $ of the starting basis set
x
=
$C
(-45)
For our purposes, the elements of $ will be chosen to be the molecular orbitals of the donor and acceptor in the complex, the and the A. If eq A5 is substituted into eq A4 and the result multiplied from the left by $+ and integration carried out over all space, we have
+
FC
= SCe
(-46)
where F is the Fock matrix with respect to the starting basis set and S is the overlap matrix. This has nontrivial solutions if and only if
F-ES=O
The one-electron solutions to eq A8 will have the form
+ CItXi CIIa+a + C I I A ~
XII
=i:
Sat = (+aIXJ (AW Within the context of ZDO all of the 2 integrals except those of the form Zuuuo would vanish. I n any event, for these types of systems they should be small; thus we shall neglect them for the moment. For weak complexes, C I ~nearly equals unity while C I ~nearly equals zero. If we set these equal to these values except for in the crossed term in Fa, and neglect the Z integrals mentioned above, we have
(A91
+ It, + 2Zaat'
(AW
Iai - CIaCIJaa"
(A19)
Faa
=
Ftt =
Fat
=
Iaa
(A171
Zaa""
The H o m e operator has contributions from both the donor and the acceptor. The I integrals can be broken down into their donor and acceptor components.
+ IaaA ItiD + I 2
Iua = Iaa
(A71
Equation A7 is analogous to the matrix notation for the usual secular determinant of LCAO theory. For the two-electron treatment of EDA interaction, the orbitals to be considered are the highest occupied molecular orbital of the donor (HOMO)Dand the lowest empty molecular orbital of the acceptor (LEMO)A, denoted +a and Xi, respectively. Expanding eq A7 in terms of the resulting 2 x 2 determinant, we have
XI = C I a + a
where w is a generalized member of the basis set and u,
v, w, and x are generalized indices ( i e . , wa would equal xu, etc.). Also
It,
=
D
(A20) (-421)
The term IaaAi s very small for a neutral acceptor molecule, while the term IiiD is primarily electrostatic in nature. Thus we have for Fa, and F,,
Fa, S Iaa D Fit
+
s litA +litD
zaaaa
+ 22auit
(A23)
Within the context of Koopmans' t h e ~ r e meq , ~ A22 is just the negative of the ionization potential of the donor, while the electron affinity of the acceptor is just -IttA, The remaining terms in eq A23 are electrostatic in nature. Thus, for weak complexes, evoking ZDO and the other mentioned approximations, eq A8 may be expressed
(A10)
Constructing the F matrix elements of eq A8 in terms
This i s just the previously used form for neutral donors and acceptors. To adequately account for electron interaction in stronger complexes, the Fock determinant of eq A8 should be used, and iteration to selfconsistency should be carried out. The generalization to a polyelectron, polyorbital treatment based on the molecular orbital formalism is straightforward. The dimensions of the F matrix will be the dimensions of the basis set. For the most general case, this will consist of all of the occupied and unoccupied molecular orbitals of the donor and the acceptor. The individual F matrix elements for a closed shell system will be of the general form Fuu
Iuu
+
occ
n=l
w,z
Cnmcnw[2zuuwz-
zu2m01
(A25)
MO THEORY OF E D A COMPLEXES
2117
where the summation index n runs over all the (doubly) occupied complex orbitals. The energy for any state in this formalism can be obtained from the solution of the ground-state problem by 000
E
H,,,)
= n'
+
m'n'
(G,rmin'n'- 68mtntGmtntn'm') (A26)
where the summation over n' and m' is over (singly) occupied spin orbitals of the entire complex and the is a Kroneker 6 function, equalling unity if term the spins are alike and zero if they are different. The H and G integrals in terms of the individual donor and acceptor orbitals are 6Smtnj
Hntnl = CCCsuCnJuv
(A271
u v
Gmf, tn'"'C
CC,luCml vCn~,Cn~xZ, vwx
(A28)
u v w x
B. Valence Bond Formalism. Equation A24 can also be derived from a polyelectron valence bond formalism where the orbitals involved are again the molecular orbitals of the donor and acceptor. The results are just a polyelectron expansion of Mulliken's treatment of electron donor-acceptor interactions. l9 For the two-electron E D A interaction the ground- and first excited singlet configurations are represented by the determinantal wave functions XI and XII XI
= det { $a(1) $a(2>
1
(A30)
SI 11 =
(A38)
Sat
with the subsidiary definitions previously given in eq A14-Al6. Substituting in for the ionization potential of the donor and the electron affinity of the acceptor, we have
+ Iaa E - ( E . A . ) A + Iaa - 2IatSat + IttD + + HI I S -(I.€'.)D
HII 11
(A39)
(-440) Subtracting I,, from both of these, redefining the last three terms in eq 40 as V,, and eq A37 as PDA, and evoking the zero differential overlap approximation again yields eq A24 for the secular determinant to be solved. The polyelectron valence bond treatment is similar to the two-electron treatment. If only the simple first CT transition is to be considered, the determinantal equation to be solved is again eq A34. The determinantal wave functions for the ground- and first charge-transfer configurations are (considering an m electron donor and an n electron acceptor, both with closed shell ground states) as in eq A41 and A42. Zaa"
ztaat
XI = det [ $ 1 ( 1 ) $ 1 ( 2 ) -+,lz(m - l)$m/z(m)xi(m l)Xi(m L/z(m n - l ) L / z ( m
+ 2 ) --+ + n)l (-441) XII l / f i det[+1(1)$1(2) -+m/z(m- 1)L/z+i(m)xi(m+ 1)Xl(m + 2 ) --+ n - l)Xn/z(m+ n ) ] det[$1(1)$1(2) --Xm/z+i(m- l)$m,z(m)h(m,+ l ) h ( m + 2 ) h z ( m + n - l)Xnp(m + n) (-442) +
=
Xn/2(m
-
--
(A31) The wave function of the complex is a linear combination of X I and XII
+
X = CIXI
G432) The energy in this approximation can be obtained from an application of the variational principle to eq A33 where Hop represents the (two-electron) Hamiltonian for the entire system and E is the energy within this approximation. ~11x11
This yields the secular determinant
I
HII-E HII I
- SIIIE
I1
-
HII 11 - E
1
=
0 (A34)
where NAB and XAB have their usual meanings. I n terms of integrals over molecular orbitals, these elements can be expressed as in
The lir matrix elements from these are more complex than eq A35-A38. They can, however be constructed in the usual manner for valence bond functions.20 An empirical interpretation of the resulting functions, however, again leads to eq A24 as the secular determinant to be solved. Higher charge-transfer interactions, "reverse dative" interactions,a and local excitations within the donor and acceptor can be handled straightforwardly within the polyelectron valence bond treatment. The polyelectron treatment based on the molecular orbital formalism is simpler to carry out in that only a single determinantal wave function need be used. All of the energy states can be approximated with the resulting wave function. These will be less accurate than the complete configuration interaction treatment based on the valence bond formalism; however, these can be improved by carrying out configuration interaction within the molecular orbital formalism. (19) R. S. Mulliken, J. Amer. Chem. Soc., 74, 811 (1952). (20) L. Pauling and E. B. Wilson, Jr., "Introduction to Quantum Mechanics," McGraw-Hill Book Co., Inc., New York, N. Y., 1968, Chapter XIII.
Volume 79, Number 7 Julv 1969
