1508
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
J. D. Doll and W . P. Reinhardt
mensions or to more chemically interesting collision partners much more feasible.
References and Notes (1) (2) (3) (4)
B. C. Garrett and W. H. Miller, J . Chem. Phys., 68, 4051 (1978). W. H. Miller, J . Chem. Phys., 50, 407 (1969). J. W. Duff and D. G. Truhlar, Chem. Phys. Lett., 23, 327 (1973). See,for example, M. S. Child, “Molecular Collision Theory”, Academic Press, New York, 1974, pp 40-43. (5) See, for example, E. T. Whittaker and G. N. Watson, “Modern Analvsis”, 4th ed, Cambridae Universitv Press. London. 1973. D 19. (6) R. N. Porter and M. Karphs, J . Chem. Phys., 40, 1105 (1964). (7) See, for example, L. I. Schiff, “Quantum Mechanics”, 3rd ed, McGraw-Hill, New York, 1968, pp 71-72. Schiff uses the parameter (Y rather than our p and sets R, = 0.
Acknowledgment. This work has been supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, U.S. Department of Energy, and all calculations were carried out on a Harris Slash Four minicomwter funded by a Science Foundation Grant 7622621. W.H.M. also acknowledges support of the Miller Institute of Basic Research in Science.
7r
Electron Theories Viewed as Parametrizations of the One-Body Green’s Function* Jimmle D. Dollt Department of Chemistry, State University of New York, Stony Brook, New York 11794
and William P. Reinhardt * Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138, and Department of Chemistry, University of Colorado, and Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado, Boulder, Colorado 80309 (Received January 2, 1979) Publication costs assisted by the National Science Foundation
Rather than viewing semiempirical a-electron theories in terms of a parametrized Hartree-Fock level theory, or in terms of valence space matrix elements of an effective many-electron a Hamiltonian, we consider a Huckel-like theory arising in the context of the theory of the one-electron field theoretic Green’s function. The one-electron Green’s function allows exact evaluation of many of the electronic properties of a molecular species and its neighboring positive and negative ions. Among these are the ionization potential, electron affinity, first-order reduced density matrix, total electronic energy, and the electronic spectra of the neighboring ions. As these quantities are given in terms of the one-body operator Z,(E), an orbital-like one-body parametrization is easily set up. For planar systems the u and a one-body spaces formally decouple, allowing an exact a-electron theory to be developed. Parametrization of polyenes and alternate hydrocarbons is discussed, as are limitations on the transferability of parameters. The possibility of expressing the total electronic energy as a sum of orbital-like energies is discussed, and it is seen that the Huckel-like parameters must be modified if this is to be done in a reasonable manner.
I. Introduction While the Huckel theory may be derived on purely heuristic grounds, both it and the more elaborate T electron theories, such as that of Pariser-Parr-Pople (PPP, hereafter), are usually discussed in terms of one-electron orbital pictures in the SCF approximation, and in terms of a static o-a separability assumpti0n.l Parameters in these models are often2 obtained by analysis of experimental data, and thus are presumed to provide more information than is contained in the exact solution of the Hartree-Fock theoretical framework that generated them. For the Huckel model in particular, the origin and interpretation of this additional information is vague, and as a result parameters are dependent on which experimental data are used to obtain them.l However, these semiempirical a-electron models do work extraordinarily (Based in part on J. D.Doll, Thesis, Harvard University, Cambridge, Mass., 1971 (unpublished). $Alfred P. Sloan Fellow (1976-1978). * J. S. Guggenheim Memorial Fellow (1978),University of Colorado Council on Research and Creative Work Faculty Fellow (1978). Visiting Professor at H m w d University, fall 1978. Permanent address, University of Colorado. 0022-3654/79/2083-1508$01 .OO/O
well for individual properties within classes of similar molecules, and certainly form the basis of much of the quantum mechanical intuition of many chemists. One would thus like to understand both the origins and successes of these models: within a general theoretical framework, so that the nature of the approximations involved can be studied and evaluated. An approach to this problem is to define an effective a Hamiltonian that allows the a electrons to correlate with one another explicitly, and with the o-core implicitly. Linderberg and Ohm3 have given a derivation of a PPP-like 7r Hamiltonian starting from the full, second quantized Hamiltonian, and following the Lykos-Parr4 analysis of U-T separability, obtaining an approximate ~ within the effective x Hamiltonian. H a r r i ~ ,again framework of second quantization, has derived a more general effective a Hamiltonian, showing how approximate core polarization and seli-consistent screening may be built into the model. Terms that allow for ‘‘nOnconSerVation’’ Iwata of x electrons are omitted. Freed,6 Westhaus et and Freed,* and others; have carried this one step further. Using a cluster ana1ysis,l0these authors have obtained an effective a Hamiltonian and ab initio values of the ‘‘true” semiempirical parameters. This work has recently been 0 1979 American
Chemical Society
Parametrizations of the One-Body Green’s Function
reviewed and analyzed by Brandowl‘ within the framework of open shell effective operators, and linked Rayleigh Schrodinger diagrammatic expansions.12 An alternative approach, which we develop here,13is to view the Huckel and PPP models as parametrizations of an exact “one-electron” theory, rather than in terms of the derivation of an effective many-electron a Hami1t0nian.l~ This approach preserves the intuitive “orbital” basis of the description as long as possible, and gives a clear indication of when one might expect difficulties. It also suggests why different experimental properties require different Huckel parameters, even for a single molecule. The extended “one-electron” theory to be parametrized is that of the one-particle mlany-body Green’s function G(rt,r’t 9 which describes the motion of a “particle” or a “hole” moving in a fully correlated fashion through an N-body ~ y s t e m . ’ ~ J ~ Knowledge of IG(rt,r’t9,or its equivalent transforms, yields exact, fully correlated values for the following: the oneelectron density matrix, and thus the natural orbitals; the ionization potential and electron affinity; the electronic spectra of the N f 1 particle positive and negative ions and the ground state energy of the original N-body reference system. Thus the parametrizations presented here will not, for examplle, give the spectrum of the N-body system itself. However, enough information is contained in G(rt,r’,t? to warrant its investigation. The plan of the paper is as follows: Elements of the many-body Green’s function formalism are reviewed in section 11, followed, in section 111, by a discussion of a Huckel-like parametrization of ethylene and benzene in terms of energy dependent CY and fi parameters. Explicit recipes are given for the perturbative calculation of these parameters, which are seen to contain effects not only of Coulomb and exchange interactions, but also a-a and u-T correlation and screening, and cr--K nonconservation. Application is made to the ionization potentials of condensed aromatics, where it is argued that the energy dependence of the parameters may be neglected. In section IV the explicit energy dependence of the parameters is exiploited in a discussion of the spectra of the positive and negative ions deriving from a closed shell alternate hydrocarbon. In section V limitations on the transferability of the CY and fi parameters are discussed, and a PPP-type theory is introduced. The paper concludes, in section VI, with a brief summary.
11. Green’s Function Formalism. P l a n a r Systems To establish notation we briefly review, in the first part of this section, some of the key expressions involved in setting up the many-body Green’s function formalism, referring to the extensive literature for mathematical details and physical m o t i v a t i ~ n . ~ ~In- ~section ~ , ~ ~ 1IB specialization to planar systems is made. The fact that the one-body Green’s function separates under the simple, nondynamical assumption of planarity is the basis of the subsequent analysis. A . Preliminaries. The one-body Green’s function is defined as1* 1 G(r,r’,t-t? T ( \kN,@IT($(rt)$t(r’t ?)l\kN,O) (2.1) 1
In eq 2.1 $(rt3 and ++(r’t? are the Heisenberg field operators which obey the usual fermion anticommutation relations, is the (exact) wave function of the ground state of the N-electroin reference system, and T i s the time ordering operator. A s discussed in standard treatments (see ref 15) knowledge of G(r,r’,t-t? yields the exact 1 matrix, yN, for the N-electron ground state, the exact ground state energy ENiOand the excitation energies of the
The Journal of Physical Chemjstty, Vol. 83, No. 11, 1979
1509
neighboring N f 1electron systems. This latter is perhaps not surprising as for times between t and t’, G(r,r’,t-t? samples the dynamics of the N f 1 electron systems obtained by the sudden addition or subtraction of one electron; the spectra of the N f 1electron ions appear in the Fourier transform of the time-dependent response to this strong perturbation. In applications the time transformed Green’s function G(r,r’,E) or its matrix representation G(E),19 are the quantities of practical utility. G(E) has the spectral resolution15J6
where the +.,(EJ and Ei(E) are the solutions of the energy-dependent e i g e n p r ~ b l e m ’ ~ J ~ , ~ ~
[Ho+ Z(E)I@.,(E) = E,(E)@,(E)
(2.3)
Here Ho = K E + U consists of the kinetic energy and nuclear attraction terms, and X(E)15J6,21 is the nonlocal, energy-dependent one-electron effective potential describing the motion of an electron or hole correlating with the N-electron reference state. The poles of G(E), arising as solutions of the implicit equation20
E = Ei(E) (2.4) will represent excitation energies of the form (EN*lJ- EN”), giving the spectrum of the neighboring ionic systems. The first-order reduced density matrix, yN, and the exact ground state energy EN$@ may be obtained in terms of appropriate integral representations:
+
( K E U + z ) G(z) dz (2.6) 2ai 2 where the contour @ in eq 2.5 and 2.6 encloses only those singularities of G(z) of the form In summary, the information in G(r,r’,E) may be obtained once an approximate self-energy, Z(r,r’,E),which acts as an effective one-body potential, is calculated or guessed. We note that unlike the usual LowdinZ2or F e ~ h b a c hpartitioning ~~ schemes, the effective potential, Z ( E ) ,acts only within the space of one particle functions, rather than in the space of N-body antisymmetric wave functions. We also note that no approximations have been made at this point, other than the standard assumptions of a nonrelativistic Born-Oppenheimer model. B. Specialization to Planar Systems. The *-electron work of Linderberg and Ohrn (L-O),3 and of H a r r i ~ , ~ begins with the decomposition of the field operators into cr and R components. For a system with a reflection plane containing all of the nuclei, and assuming a Born-Oppenheimer picture, this separation is exact:
E“,@=
-Tr$
+ $,(rt) $+W= $,+(rt) + $,+kt1 $(rt) = $Art)
(2.7a) (2.7b)
where the cr and T electrons created (annihilated) have the usual “cr” and ‘‘T” symmetries. Substitution of eq 2.7a,b into the Hamiltonian 7f = $dr$+(rt)(KE
+ U)$(rt) +
unfortunately does not uncouple the system into the sum of a u and a Hamiltonians.
1510
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
It is instructive to see why this is so. The field operators transform under the reflection operation (with respect to the nuclear plane) as R$,(rt) = +Art) (2.9a) R+,(rt) = -+Art)
(2.9b)
and similarly for $,+, $,+. As the electronic Hamiltonian is invariant to such transformations, we conclude that there must be an even number of x operators in each term of 54, Thus, we have at once J d r $+(rt)HO(r)+(rt) = J d r $,+(rt)HO(r)$,(rt)
+
Jdr
$,+WWr)$,W (2.10)
giving rise to one-body u-x separability. The two-body contribution (2.11) gives rise (as may be easily checked) to eight nonvanishing terms, one of which contains only u field operators, one only x operators. The other six terms each involve u-x coupling, and each contain two u field operators and two x field operators. L 1 0 and Harris have analyzed these coupling terms. L-O3 derive the PPP Hamiltonian in second quantized form, and Harris5 sets up a self-consistent, dielectrically screened, effective x-space Hamiltonian using a van Vleck transformation, but neglecting terms that do not conserve the number of u and x electrons. We note at once the formal advantage of treating planar systems within the framework of the one-particle Green’s function: ( \kN,OIT[$(rt)rl,+(r’~?]l\kN,O) = (\kN~oIT[$o(rt)$o(r’t’)] X /\kN,O) ( \kN,OIT[$,(rt)$,+(r’t?]I\kN,O) (2.12)
J. D. Doll and W. P. Reinhardt
This implies that the a total energy may be computed from knowledge of G, alone, and it follows that Etotal
= E,h)
+ E,(u)
(2.14e)
which is the exact generalization of Lykos and Parr’s4 theorems 2 and 3. The a-electron ionization potentials (1P) and electron affinities (EA), and electronic excitations of the neighboring N f 1 electron x systems may be computed from G, alone; and Y,, the x-electron 1 matrix, may be computed from G, alone. We stress again that no approximations have yet been made. The separation of the Green’s function into x and u parts follows from the fact that it is in one-particle rather than a many particle object. 111. Huckel Parametrization A . Formal Parametrization. The simple existence of the one-electron effective Hamiltonian (KE v), Z,(E) immediately allows a Huckel type parametrization to be carried out with a meaningful parameter identification. More usual derivations are often misleading, as the basic operators and matrix elements are not well defined, leading, for example, to the erroneous conclusion that the Huckel cy parameter neglects electron repu1sion.l In order to obtain an exact x-electron parametrization for a system such as ethylene within the valence space, the x space itself must be partitioned into valence and nonvalence MO’s. This technical, but straightforward, trick is discussed in the Appendix, tpd leafis to an effective n-electron valence space operator (KE v), + Z,(E), which has only r-electron valence space matrix elements, yet contains the effects of excited a states as well as the implicit u dynamics. We now consider ethylene as a simple example. Working in the primitive atomic valence R basis of 2pa orbitals, (labeled 1,2) on each carbon (see Appendix), the effective x problem for ethylene is
+
+
+
+
as the one-body expectation value is reflection invariant. Thus G(r,r’,t-t ’) = G,,(r,r’,t-t’) + G,(r,r’,t-t? (2.13a)
where (again, see Appendix)
and
B(E) =
G(r,r’,E) = G,(r,r’,E)
+ G,(r,r’,E)
(2.13b)
with no approximation, other than the assumption of planarity. Thus, the one-body Green’s function separates exactly into u and x parts with no assumptions about the dynamics of the u and x systems. It thus makes sense to consider only the x one-body Green’s function G,, depending explicitly on the x electrons and implicitly on the u electrons, as an independent object, and we base our semiempirical analysis on it. We note that the decoupling of eq 2.13a,b also implies that Z ( E ) is diagonal in the c, a indicies, as follows from the fact that it is a one-body effective operator, and must conserve the reflection parity. In summary, we have G(E) = GJE) + G,(E) (2.14a) Z,,,(E) = Z , , m = 0
(2.14b)
and thus for example G,(E) = { E - [ffo + 2WIT,J1
(2.14~)
and
6
E,(u) = - Tr 2xi 2
(KE + U
+ z) GJz)
dz
(2.14d)
a(E)=
(I?/ + 5,(E))11 = (272+ 5,(E))22 (ATo + &(E))l2= (I?: + 5,(E))21 s 1 2
=
s 2 1=
(41,42)
(3.2a) (3.2b) (3.2~)
assuming real orbitals. We have written A2 = RE, + UT, As no approximations have been made, we can conclude that the Huckel-like parametrization of eq 3.1 contains screening, electron repulsion, correlation, and u-x interactions. The eigenvalues of the Hamiltonian of eq 3.1 are (3.3) where X,is the ith eigenvalue of the Huckel topological matrix.24 Ionization potentials and electron affinities are obtained as roots of the implicit equation16,z0 E = E,(E) (3.4) As these roots are singularities of G,(E) they may be unambiguously identified as IP’s or EA’S,without recourse to Koopman‘s theorem. If, as will be made plausible in sections IIIC and IIID, the energy dependence of a ( E ) and P(E) is not strong over the range between the IP and EA we can assume that values of the two parameters cy and p can be determined from the vertical IP and EA of ethylene, and that such values would be those obtained
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
Parametrizations of the One-Body Green’s Function
i
by actual ab initio calculation of the parameters via eq 3.2a,b. Thus, for the very simple case at hand we see that the Huckel parametrization is an exact parametrization of an exact effective x-electron one-particle operator. The details of how this simple parametrization contains the expected physical effects will become evident in section IIIB. Extension of the above ideas to more complex systems does require a more extensive analysis. For example for benzene, the ,Jr-electron effective Hamiltonian is given as
lY,,= .,,(E)
=
(82+ 3 J E ) ) , ,
i = 1, 2, ,.., 6 H,] = &](E)= (Q + 3,(E))bl i # j = 1, 2, ..., 6 sl]
=
($1741)
requiring solution of the secular problem [ E 5 - K ( E ) ] c= 0
A
B
C
D
(3.5a) (3.5b) (3.5c) (3.6)
to construct the orbitals and eigenenergies. If we assume that 6, = S, == 0 for i and j not neighbors, that is that the 6,and S, are local quantities not depending on “far away” carbons, we regain the form of the Huckel approximation, but within a framework that allows the possibility of ab initio evaluation of the validity of the a p p r o ~ i m a t i o n .In ~~ more general, nonsymmetric cases the theoretical a’s and p’s will depend on their location within the molecule, but again in a way that can be calculated from first principles. B. Perturbatwe Expansions of Formal Huckel Parameters. In this section we examine the perturbative expression of C,(E), and give explicit recipes for construction of low order contributions to a ( E ) and P(E). Interpretation of the perturbative contributions to the self-energy gives an expicit indication of the physical effects summarized by the parametrization. As discussed in the Appendix, 2,(E) is constructed in an MO basis within the valence space. The Huckel parameters are obtained from projection of this basis onto the individual atomic 2px orbitals 1hat define the valence space. To give a feeling for the physical effects included in the a’s and p’s, we assume that H,O is the interaction of a free x electron with ithe nuclear frame. We note that in an actual calculation, one would more likely begin with HartreeFock x orbitals, incorporating substantial static and exchange screening of the nuclei. The development here is perturbative, f;llowing standard rules for the perturbative expansion of Zeroth-Order Contributions.26 a,,O = (ilH,Oli)
(3.7a)
PlIo = (ilH21j) i # j , but bonded Pllo = 0 i = j , i j nonbonded
(3.7b) (3.7c)
The a,: and P,: defined here will be sensitive to the whole molecular frame owing to the long-range Coulomb interaction U(r);one cannot expect molecule independence or transferabiility of these zeroth order quantities. First-Order Contributions. The first-order contributions to a and 0are given in terms of the diagrams of Figure 1. These correspond to the matrix elements
[%W)Icl
1511
(iI%(l)(E)Ii)= C[(iu,lIju,) - ( i u , l l u ~ ) ] + UP
~ [ ( i x , l l . h , ) - (ix,lIx$)l (3.8) =Y
which give a l l Lif i = j and 6,‘ for i # j , x-bonded atoms.
Flgure 1. Diagrammatic representation of the first-order contributions to the a and p Huckel parameters. Diagram A is the direct Coulomb repulsion interaction with the charge density of the pth occupied u-type MO. Diagram B is the corresponding exchange interaction. Diagrams C and D are the direct exchange interactions with occupied 7r-type MO’s.
i
i
A
B
C
D
Figure 2. Secondarder contributions to the a and fl Huckel parameters. Diagrams A, B, and C correspond to the second-order irreducible perturbative contributions to the self-energy. The corresponding exchange diagrams are not shown. Diagram D arises from the partitioning discussed in the Appendix. Diagrams A, B, and C give energy-dependent contributions to a and fl resulting from dynamic polarization of the u core and from x-T correlation.
These terms represent the direct and exchange Coulombic interaction with u and x electrons on each of the nuclei. It is this “screening” of far away nuclei that cancels, at least in part, the long-range contributions to a,: and 0,;. Second-Order Contributions. Second-order contributions to a and P are given in terms of the diagrams of Figure 2. The diagram of Figure 2D, which appears to be a “reducible” contribution,15arises from the partitioning, and subsequent projection into the valence subspace as discussed in the Appendix. Diagram 2A (and its exchange) is a x-x correlation term contributing to a double x,x, xpxqexcitation in CI language. The diagram of Figure 2B represents polarization of the u core with a subsequent virtual x excitation. In CI language it is a component of the double excitation u,xv The diagram of Figure 2C is a “nonconserving” contribution to the parameters, and has the double excitation x P x , upuqas an intuitive
-
-
-
1512
The Journal of Physical Chemistry, Vol. 83, No. 11,
J. D. Doll and W. P. Reinhardt
1979 -9.30 -8.90 -8.50
-6,90y ,__,
, ,
LINEAR FIT QUADRATIC FIT X X X E X P E R I M E N T A L POINTS
-6.50
X -6.10
ENERGY
0.20 0.30 0.40 0 . 5 0
Figure 3. Plot of the roots E,(€) as a function of €for a hypothetical 3 X 3 matrix representation of G(€). Poles of G(E)are at the solutions of the implicit equation € = €,(E), and are labeled 1 through 8. The roots are of two distinct types: Those near “0” (Le., 3, 4, 5) corresponding to the three eigenvalues of a nearly energy-independent effective Hamiltonian, and those (1, 2 , 6, 7, 8) induced by poles of the As indicated in the figure, for roots 3 and 4 self-energy d€,(E)ld€l,=,, = 0 as IStypical for this type of root, while d€,(E)/d€,=,, is large and negative for the induced roots. As discussed in the text, these conditions on the derivative are related to the goodness of the orbital description, and to the possibility of writing the total electronic energy as the sum of N orbital-like energies.
E(€).
counterpart. The second-order contribution to a,, and p, is given in terms of
[ z , ( 2 ) ( E ) ]=, (lzT@)(E)1j) = (ixpI1 xvx,,)[ ( xvxplljxp)- (x,x,/IxJ (E + t p - t, - tu) xrx.xp
c
c
(ix,ll
XpXqII
)Kxpxqlljx,)
-
(xpxqllxJ)l
)I
+
+
( E + t,, - ep - tq) energy independent second-order diagrams [all of which are absent in a Hartree-Fock level treatment] (3.9) XirXpXq
In eq 3.8 and 3.9 the 11 notation implies a Coulomb integral, and x is a u or x M O with subscripts ~ , indicating v occupied MO’s in a single determinantal approximation. The p,q refer to unoccupied (i.e., excited) orbitals. Higher order contributions may be obtained from the usual expansion of z ( E ) . However, these second-order terms are sufficient to indicate that n-correlation, u-polarization, exchange, and correlated u-x interactions do contribute to the parameters. C. Energy Dependence of the Parameters. The second-order contributions to CY and are explicitly energy dependent, as is seen in eq 3.9. In fact, the terms shown in eq 3.9 have poles (Le., singularities) at the energies e,, + E , - E,, and tp + tq - e,,. Thus the energy dependence does not even appear, a t first glance, to be weak. However, this is deceiving, as will be discussed in detail in section IIIE. In this section we give a brief, qualitative, discussion of the energy dependences in the region of the EA and IP, in preparation for a discussion of Huckel parametrizations of IP’s in section IIID. The role of the poles in the contributions to the selfenergy is to induce singularities in G(r,r’,E) corresponding to ionization followed by excitation of the resulting N 1body system, or to addition of an electron plus excitation of the resulting N + 1 body system. Indeed, the poles of Z ( E ) occur at zero-order approximations to these energies.
0.60 0.70 0.80 0.90
1.00
X Figure 4. Linear and quadratic fits of the ionization potentials of condensed aromatic hydrocarbons (see Table I) vs. the Huckel structure factor 5,. The quadratic fit gives only a moderately improved rms deviation, but clearly gives a better overall fR, resulting in the parameters discussed in section HID.
This is illustrated in Figure 3 where the root structure of the implicit equations E,@) = E n = 1, 2 , 3 (3.10) is displayed for a 3 x 3 model problem. The eight roots shown all correspond to poles of G(E). These roots are seen to arise in two distinct manners: Roots 1, 2 , 6, 7, and 8 appear only because they are induced by the poles of Z ( E ) a t energies A , B , C, D, and E. Roots 3, 4, and 5 correspond more simply and directly to the filled and unfilled MO’s of the problsm. Near the “zero” of energy, away from the poles of Z ( E ) , the E,(E) are relatively constant. In particular near the roots E,(E) = E (labeled as root 3) (3.11a) corresponding to the IP, and E,(E) = E (labeled as root 4)
(3.11b)
corresponding to the EA, there is very little energy dependence. TJis suggests an energy independent parametrization of Z(E) over this limited energy range. This will be the case unless there are low-lying excited electronic states of the neighboring N f 1 particle systems. Thus, again referring to Figure 3, root 5 would not be well approximated using an energy independent parameterization suitable for roots 3 and 4, which are farther away from the poles of z ( E ) . These ideas are made quantitative in section IIIE. D. Huckel Parametrization of I P S . Assuming that the energy dependence of the parameters may be neglected in the region of an IP, we can test the approximations of section IIIB by evaluating of a and p experimentally. Within the valence we shall have (3.12) where lk) is the kth eigenvector of the topological matrix with eigenvalue Xk a
+ x,p
E , = 1 + XRS’
(3.13)
where S’is the nearest neighbor overlap. Examination of the explicit expressions for a and 0 indicates an implicit
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
Parametrizations of the One-Body Green’s Function
1513
TABLE I: Average Bond Orders p, Huckel Structure Factors Xi,and Experimentala IP’s Compared with Values Obtained with Linearb and QuadraticC Determination of the Parameters, as Discussed in Section IIID’ compd
P
Xi
IPb (expt)
IPc (calcd)
Ipd (calcd)
0.587
0.220
6.23
6.45
6.27
0.594
0.294
6.64
6.74
6.65
0.588
0.347
6.83
6.93
6.89
0.604
0.414
7.23
7.19
7.20
0.592
0.437
7.35
7.28
7.30
0.598
0.452
7.42
7.34
7.36
0.594
0.473
7.42
7.41
7.45
0.594
0.492
7.42
7.47
7.53
0.595
0.499
7.43
7.51
7.56
0.592
0.445
7.58
7.30
7.33
0.595
0.502
7.62
7.52
7.57
0.600
0.520
7.72
7.59
7.65
0.600
0.568
7.76
7.71
7.83
0.622
0.618
8.12
7.96
8.03
0.602
0.6 8 4
8.13
8.21
8.21
0.667
1.000
9.25
9.41
9.22
u u
The IP as a function of X is shown in Figure 4 for the linear and quadratic fits. Data from R. Dandel, R. Lefebure, and and C. Moser, “Quantum Chemistry, Method and Applications”, Interscience, New York, 1959. Linear least-squares fit. Z(expt - calcd)2 = 0.26. Quadratic least-squares fit. ~ ( e x p-t calcd)’ = 0.15.
dependence, through the filled and unfilled MO’s, on the charge densities and bond orders. We would thus expect a consistent set of a , P’s to follow only from a closely
related series of compounds. This is exemplified by the analysis shown in Table I and in Figure 4. The first-order expansion of eq 3.13
1514
The Journal of Physical Chemistry, Vol. 83, No. 11,
E, = a
+ X,(p
-
as’)
1979
(3.14)
gives the linear least-squares fit of the IP’s of a closely related set of condensed aromatic ring systems, yielding (see Figure 4) a = -5.63 eV (3.15a)
(0- as? = -3.78
eV
(3.15b)
J. D. Doll and W. P. Reinhardt
propriate poles, ? j is defined by eq 3.20, and where (3.21) is a term arising from the fact that poles of G(z) are not simple. In a similar way the r-electron 1 matrix is given as
and an rms deviation of -0.03 eV. The quadratic expansion
E, = a
+ ( p - &’)Xi + [a(S?Z psqx,z
(3.15~)
-
gives a slightly better quantitative fit (rms deviation -0.015 eV), but one that better represents the extreme X,values, and a = -5.07 eV (3.15d)
p = -7.87 eV
(3.15e)
S’= 0.29
(3.15f)
We note that the empirically obtained next neighbor overlap, S’ = 0.29, is in good agreement with the value of 1/4 usually assumed to represent a “typical” overlap. The fit is not, of course, exact, but is better than could be expected from a simple application of Koopman’s theorem to the minimal basis SCF results which are often thought to be those represented by the Huckel model. It would be a challenge to reproduce these values of a and p from ab initio calculations of the self-energy. E. Total Energies and Goodness of the Orbital Description. There has been a recent flurry of interest in the relationship between the sum of Hartree-Fock orbital energies
P
-
,V
1
(3.17)
where the contour encloses all singularities of G ( z )of the type (ENso- Eh”lJ),j = 0, 1, 2, ..., but only those singularities. In principle, this is an infinite sum, and it would seem difficult to reduce it to the sum of N terms in any reasonable fashion. However, in terms of the energy dependent eigenvalues E,(E), and eigenfunctions a n ( E )
[Ho + Z(E)]@,(E)= E,(E)@JE)
(3.23)
where, again
and actual total electronic energies.27 Within the framework of the one-body Green’s function theory, we can ask under what circumstances might it be reasonable to approximate an exact correlated N-body total energy by the sum of N “orbital-like’’ energies, and can this be done with an energy independent one-body parametrization? The electronic energy of the r electrons is given in terms of the contour integral
( K E + U + z ) G(z) dz
cz1
c=1
(3.16)
61
(3.22)
where again the sum runs over j = 1 , 2 , 3 , ... corresponding to poles of G(z) at each of the energies ENro- EN-lJ. Equations 3.20 and 3.22 can, perhaps surprisingly, be reduced to rather simple expressions when the orbital approximation is valid as a good zero-order description of the system. This follows from the fact that the gl’s are of two types, as illustrated in Figure 3. For a pole of G ( z ) corresponding (approximately) to an unperturbed orbital energy IdEn(E)/dEl > 1, and as dE,(E)/dE is always negative, we have gl
