2288
J. DE HEER
Vol. 66
A REFINED ALTERNANT MOLECULAR ORBITAL TREATMENT OF THE GROUND STATE OF BENZENE BY J. DE HEER Department of Chemistry, University of Colorado, Boulder, Colorado Received May $8,1966
In the refined alternant molecular orbital (AMO) method we allow for different variable parameters for each M O pair which is coupled. Application to the ground state of benzene reveals this method to be extremely powerful: The A M 0 wave function, involving two variable parameters only, has a lower energy than the nine-term configuration interaction (CI) wave function which was obtained by Parr, Craig, and Ross. A further numerical analysis reveals that the relative improvement of the refined over the simple method is independent of any reasonable choice of core potential and electron interaction integral approximation. 9 detailed comparison of the A M 0 and CI wave functions is given.
Introduction Dewar’ has discussed a way in which to t’reatthe “vertical correlation” in conjugated systems. The alternant molecular orbital (AMO) method concerns itself with t,he more conventional “horizont.al correlation” which, in quantum chemistry, usually is treated by means of configuration interaction (CI) t8echniques.2 As such, the AM0 method is by no means confined to a-electrons; it pertains to all systems of alternant character, ie., consisting of a set of identical atomic centers which can be divided into two sub-sets in such a way that no two atoms of the same sub-set are neighbors t,o each other. In fact, when Lowdin proposed the use of alternant molecular orbitals to treat electron ~orrelation,~ he primarily envisaged applications to solid state problems, and it is quite possible that here they may ultimately find their most useful domain. To date, however, the more interesting applicat’ionshave been in t.he field of conjugated system^^-^ and it appears appropriate to give a first application of a refined A M 0 method within this framework. More specifically, we shall treat the ground stat’e of the a-electron sext,et in benzene. Not only has this been the init’ialtest case for the A M 0 method in it’s original form,4 but it also has been the subject of a most extensive CI treatment,’so that all relevant data are available for comparison. The Simple and Refined AM0 Method.The effectiveness of the AM0 method is due to the fact that a major portion of the correlation between electrons with different spins can be taken into account by assigning these t’odifferent, “semilocalized” orbitals so that they can avoid each other as much as possible.* In alternant systems, this is achieved by taking suitable linear combinations of sets of (‘paired” bonding and ant,ibonding MO, !bk and $i;>to yield the AMO, U K and E K , thus (1) M. J. S. Dewar and N. L. Sabelli, J . Phzls. Chem., 66, 2310 (1962). (2) For a review of the correlation problem in quantum chemistry, see P. 0.Lowdin, Advan. Chem. Phye., 2, 209 (1959). (3) P. 0. Lowdin, “Symposium on Molecular Physics,” Nikko, Japan, 1953,p. 13; Phys. Rev., 97,1509 (1955). (4) T.Itoh and H. Yoahizumi, J . Phys. SOC.Japan, 10,201 (1955). (5) R. Lefebvre, H . H. Dearman, and H. M. McConnell, J. Chem. Phys.. 52, 176 (1960). (6) R. Pauncz, J. de Heer, and P. 0. Lowdin, ibid., 56, 2257 (1962). (7) R. G. Parr, D. P. Craig, and I. G. Rosa, ibid., 18, 1581 (1950). (8) Electrons with the same spin are correlated through the Pauli antisymmetrization principle. See, e.p., P. 0.Lowdin, ref. 2.
a9 = cos (I.K =
f sin 6k$&
ek$k
(1)
cos ek$k - sin &$i;
with 0
< 6k < a/4
[For a system with 2n equivalent atomic centers and 2n electrons, h: = 1, 2, . . . . . .n, hence K = I, 11, . . . , . .N . ] -4s illustrated schematically in Fig. 1, the U K and 8~ have the desired semilocalized property. The Ah10 wale function takes the form
i*o(el, . . ., e,)
*
O ~ ~ C P (a) T ~ where Boo is the projection operator selecting the proper singlet spin state 5
=
a is the antisymmetrization operator a =
= [(2n)!]-”’
(-1)PP
(4)
+ 1).. . .a,(2n)
(5)
P
aI(1). . . .us(n)cTI(n
and
+
. . . @(n)@(n1).. . .@(an) (6) To c~(1). Thus, in a loose way of speaking, we have assigned all electrons with a spin to the U K and all those with /3 spin to the E K . It is important to note that unless all 6k equal a1-4,the AM0 do not form an orthonormal set; n-hile JaK*(l)aL(l)
dci
=
J~TK*(~)(I.L(I) dol =
~ K L
(74 one also has J ~ K * ( ~ ) c ? L ( ~ )dul = X k s K L
(7b)
with x k = COS 2ek (8) As in our earlier w ~ r k ,we ~ ,shall ~ find it convenient to carry out all algebraical manipulations in terms of the X k rather than in terms of the Bk. The energy associated with the wave function “k0of eq. 2 is
(9) R. Pauncz, J de Heer. and P. 0. Lowdin, J . Chem. Phus., 36, 2247 (1962).
REFISEDALTERSANTMOLECULAR ORBITALTREATMEKT OF BESZEKE
Dec., 1962
‘Eo(&, . . 0,) .)
~3
E
=
x\k*Ho,9dT/J\k*\kdr
2289
(9)
Corresponding to the usual splitting of H,, into a “one-electron Hamiltonian,” hl(z), comprising z
kinetic energy and potential energy due to the 44 core,” and the electron interaction terms, e21 rij,l0 the energy E is written as the sum of a “oneelectron energy,” E l , and an electron interaction energy, E12. In what we shall henceforth refer to as the simwle A M 0 method, we impose the restriction that ‘all mixing parameters 0k (or Xk) have to take on the same value. The corresponding one-parameter energy expressions, El@),E12(0),and E @ ) ,for an arbitrary alternant system with closed-shell structure, have been given else~vhere.~The typical behavior of thecse quantities, referred to the energies of the relevant single ,4S?tIO determinant 2E, respectively), (denoted by 2G, U , and U as a function of tl, is illustrated in Fig. 2. Xote the characteristic monotonic increase in El and the decrease in Elz, resulting in a single minimum in E. For any particular system, a numerical analysis will have to yield the optimum halue of 0, eo, and the corresponding energy depression, A E r e l . The latter quantity denotes the improvement of the A M 0 energy over that of the single ASMO determinant, relative to the condition that all 6ko are equal. The ground state of the a-electron sextet in benzene has been treated in this fashion by Itoh and Yoshizumi“ and by Pauncz, de Heer, and Lowdin.6 The irelevant results will be referred to below. Both the strength and the limitations of the simple A M 0 method reside in the circumstance that only one variable parameter is used. In the rejlined A M 0 method the restriction of equal 6i is dropped and we obtain a more flexible, manyparameter wave function. General expressions for Ei(61,.. . . ,On), E12(01,.. . . ,en), and E(&,. . . . ,ea) will be given elsewhere.” For any particular system an energy minimization procedure will have to give us a set of optimum mixing parameters, Ole, 0 2 0 , . . . ..fino, and an energy depression, /AEl. The latter quantity reflects the improvement of the energy obtained with the refined AM0 wave function over that of the single ASNO determinant. While the possibility of developing the A M 0 method in this fashion has been apparent since its inception, and it is intuitively clear that a MO pair, +k and +E, will mix the stronger the smaller the energy gap between them, only actual computations can reveal whether the degree of improvement of the results justifies the complications which of necessity are introduced as the simpler scheme is abandoned. As far as we know, the present investigation is the first numerical analysis of this type. Reasons for selecting the benzene molecule to this purpose already have been given in the Introduction. Wave Function and Energy Expression for Benzene.--It is most convenient to write the benzene MO, +j, in terms of the carbon 2p,AO, u,, as
+
(10) See, e.g., P. G. Lykos, J . Phys. Chem., 66, 2324 (1962). (11) J. de Heer, J . Chern. Phys., In press.
. .
.
.
4 . . . e
.
e
e
e
. . .
Fig. 1.-Huckel-Wheland
M O and SMO.
The A M 0 will have to be of the type formally denoted by eq. 1 above. However, it is readily verified that, in order to get the required AI, symmetry for the total wave function, the two degenerate MO pairs must be transformed into AM0 with the same mixing parameter. Hence, in the special case under consideration, even although we are dealing with an electron sextet, we get a tworather than a three-parameter problem and the six AM0 become a. = cos
+ sin Bo+,
iio = cos Oolc.o
- sin
2290
J. a1 =
+ sin e12#-z
cos e12#1
81 = cos Blz#l
DE
HEER
(12)
Vol. 66
with
- sin e12#-2
hkk
.f#k*(l)hl(l)$k(l)
do1
(18)
E1 -2&
Thus, by eq. 2, '\ko(O0, B12) is a linear combination of twenty Slater determinants, which establishes the connection between the AM0 and the CI methods (see the discussion below). The desired energy expressions may be obtained either from the general equations given elsewhere" or by direct derivation using the techniques described by Pauncz, de Heer, and Lowdin.9 It is convenient to express these energies immediately with reference to those of the single ASMO determinant. Define 32 as 31
1
+
'/3(x02
+
2x12')
+
'/3(2h02~12'
+
f
h4) Xo2X1z4 (14) Then the one-electron energy becomes
-
El(Xo,X12) - E1(1,1) = A W 4 - Po(1 x12' X1z4)ho 3n.
+ +
-A-
x = o
B=O
-e-
8 = n/4
Fig. 2.-Energies
aa a function of a single mixing parameter.
The one-electron operator energies, hkk, frequently are referred to as "orbital energies" and usually are denoted by ek.4J For the electron interaction energy we obtain El2(XO,X12)
+ h12(2 + Xo2 +
x=1
- E12(1,1) =
REFIXED ALTERNANT MOLECULAR ORBITAL TREATMENT OF BENZENE
Dec., 1962
and exchange integrals, involving the MO Jli. The f i j are defined as
+i
2291
and
(22)
The use of these pij allows for a systematic notation of all electron interaction integrals needed in our investigation. For reference purposes, we give below the corresponding, partially of an ad hoc nature, symbols introduced by Parr, Craig, and ROSS,'as also used by Itoh and Yo~hizumi,~ for the four relevant integrals in benzene r1.4
Poi =
= 17:
P1,-1
Pz3lo; Pi3
=
=
E
riaa2
(23)
Using the identification (8), one may readily check that in the limiting case Xo = XIZ = X (hence 0, = e12 = e), our energy expressions reduce to the ones given by Itoh and Yoshizumi14when the latter are referred to those of the single ASMO determinant. The drastic simplifications to which their formulas can be subjected in this limiting are, on the whole, no longer applicable in the refined scheme. One further point arises with respect to eq. 19. If one uses an integral approximation in which all Coulomb integrals between MO, yij, are equal (= .y say), as; is the case if the Mulliken or ParrPariser approximation is used t o determine all electronic interaction integrals,l 2we have"
'0
I
2
3
4
5
6
7
8
9
13
-i,2The E ( X , , X , , )
Fig. 3.-The
Surface
E(A0, Xlz) surface: A, relative minimum (A, = h);0 , absolute minimum.
in their simple A M 0 study of b e n ~ e n e . ~Here oneelectron integrals mere obtained according to the original Goeppert-Mayer-Sklar approximation, two center electronic repulsion integrals were computed directly, using Slater-type orbitals, and three- and four-center integrals were evaluated with the SklarLondon approximation. Finally, we carried out a third set of calculations following the method of neglecting differential overlap, with empirically selected, much reduced, electron interaction integrals, actual values of which were taken from Pariser.'4 Our chief purpose here was to check so that, in effect, we can eliminate all yij-containing whether a correspondingly much reduced electron terms from the electron interaction energy expres- correlation would essentially alter the relative merits sion. This also shows a real advantage of comput- of the refined over the simple method. ing E12(X0,\12) - Elz(1,l) rather than En(XO,X12) While in our earlier work6 we were primarily as such. concerned with total electronic energies, it is of Numerical Analysis.-To carry out a numerical interest in the current investigation not only to analysis, we have to (i) select a specific form of the obtain E(Xo,X12),but also EI(X0,X12)and E~~(XO,XIZ) one-electron :Hamiltonian, h,, and (ii) adopt an separately. After computing the three energy approximation for the evaluation of the electron surfaces concerned over the entire range of XO interaction integrals. These matters have been and XI2, that is from zero to unity, we ascertained extensively discussed in the literaturelo and we that in each case E(Xo,X12)is simply concave. merely wish to justify our choices concerned within Figure 3 shows a projection of the latter surface on the context of the aims of the present investiga- the Xo,Xlz-planefor the results obtained with the tion. As we outlined in the Introduction, these first of the three integral approximations referred aims are twofold: to determine the improvement to above (compare the first column of numerical over the simple A M 0 method and to give a com- data in Table I). With the other two integral parison with the best available CI treatment. With approximations, the general appearance of this the former goal in mind, we have carried out com- surface remains the same. putations with the same set of integrals as used in The important numerical results are collected in our earler w 0 r k . ~ . 3 ~ That ~ 2 ~ ~is,~ we used Rueden- Table I; after each optimum 3 value the correberg's values of the one-electron integrals and a sponding 0 value is given in parentheses. combined Mulliken-tight-binding approximation Discussion for all electronic interaction integrals. To make The immediate conclusions to be drawn from the the desired comparison with the CI wave function of Parr, Craig, and ROSS,' we have repeated the data summarized in Table I are that ?ur refine calculations with the integrals used by these in- ment of the A M 0 methodr esults in a sizable imvestigators and used also by Itoh and Yoshizumi provement in energy and that the optimum values of the mixing parameters are distinctly different (12) J. de Hear and R. Paunoa, J . Md. Spectrv., 6, 326 (1960). (13) K. Ruedenberg, J . Chem. Phva., 84, 1861 (lW3l).
(14) R. Parker, ibid., 34, 250 (1956).
J. DE HEER
2292
Vol. 66
TABLE I ENERGIES A N D OPTIMUM A VALUES Mulliken-tight binding approximation"
Parr-Craig-Ross (Itoh-Yoahirumi) integrals)
Neglect of differential overlap; empirical integralsC
0.631 (25' 26') 3.993 -7.171 -3.17Sd 0.762 (20' 11') 0.510 (29'40') 4.167 -7.863 -3.696 0.518
0.691 (23'9') 2.915 -5.317 -2. 402e 0.808 (18"3') 0.590 (26' 55') 3.136 -5.928 -2.792 0.390
0.837 (16'38') 0.671 -1.236 -0.565 0.911 (120 11') 0.767 (19' 57') 0,780 -1.449 -0.669 0.104
16.27, 18.47, 16.3% AE Identical with the value obtained by a See ref. 6, 9, 12, 13. b See ref. 4, 7. 0 Numerical values from Pariser, ref. 14. Pauncz, de Heer, and Lowdin, ref. 6. e This number differa by 0.05 e.v. from the value published by Itoh and Yoshizumi, ref. 4.
TABLE I1 COMPARISON OF A M 0 A N D CI TREATMENTS Same coefficients in the nine-term C I Xo' 0 0.808; wave function of XO = O.69lb X d 0.590 Parr, Craig, Rome 0.910 0.871
Numerical --expansion coefficientsExpansion coe5cient in the A M 0 wave function"
-
01
=
+a
=
.225
.210
98
=
.130
.142
0
64
06 =
-s(X)(l
07 =
- [2]'/2
0a
- Xo)(l u(X)(l
XN)'
+ Xo)(1 - XIZ')
.052
0
0
0
06 =
0
0
-0.164
-0,093
-
-
.232
-
.318
-
.055 .os9 .227
.204
,204
.118
,069
0s = h 1 ~ ~ ) / 3 1 ~ * .134
010 =
.030
.058
Kot included
dl1
3
,042
.034
Not included
012
=
-
.024
Not included
01;
=
-
,014
X o t included
.053
Not included
.024
.030
S o t included
.006
,006
Not included
-u(X)(l
- Xo)(l
- hl2')
-
.030
dl4
'$15
s(X)(1
(W+zV-22) .(A) 7 1/4[1
U(h)(l
- X12)[2(1 - W ) ( l - h12~)/3)~/z - XO)(l - X l d *
+ l/a(Xo2 +As2x12~) + 1/~(X1~4 + 2&,*X12*) + xoBX1241 = 1/[4 d z ] . communicated to Itoh and Yoshizumi (ref. 4). and Yoshizumi (ref. 4).
6ld
a
b
The first nine already given by Itoh
6
for different MO pairs. This is true even in the case where the correlation is greatly reduced by the use of empirical two-center electron interaction
integrals14; the relative improvement in energy remains of the same order. As the result of an extensive CI calculation, in-
Dee., 1962
REFINEDALTERNAKT AIOLECULAR ORBITAL TRE.4TMEST
OF
BEXZEXE
2293
volving a nine-term wave function with nine variable parameters, Parr, Craig, and ROSS’~ obtained an energy depression, /AE~II,of 2.7 e.v. It is extremely gratifying that, with exactly the same integrals, we get an energy depression of 2.8 e.v., using two varia,ble parameters only. Independent of our choice of integral approximation, in going from the simple to the refined method, there always is a slight increase in one-electron energy, which is more than offset by an accompanying decrease in electron interaction energy. It is of interest to analyze this further by splitting up the former in its kinetic and potential energy parts. We have done this for the results obtained with the Mulliken-tight-binding approximation, where the relevant data are readily availableg.13; those with alternative integrals should give us essentially the same picture since the same underlying principles are involved. We readily compute that the 3.993 e.v. one-electron energy rise of the simple scheme is due to an increase of 11.167 e.v. in kinetic energy and a decrease of 7.624 e.v. in potential energy. When we break down the corresponding amount, 4.167 e.v.. in the refined method we get +12.124 e.v. kinetic energy and -7.957 e.v. potential energy. While the electron interaction energy depressions reflect the main portion of the correlation effect the A310 method aims at, it is no surprise that with the accompanying L‘semi-localization’’ process, one obtains a rather steep rise in kinetic energy. The sieable decrease in the potential energy part of the one-electron energy is, a t first sight, somewhat of a surprise. I t is due to the fact that as more of the antibonding MO, which have more nodes between the atoms than the bonding MO, are mixed into the wave function, the electrons are concentrated more “on’, than “between” the (positively charged) atomic centers. In going from the simple to the refined method, the kinetic energy increases by 0.507 e.v. while the potential energy decreases by 1.025 e.v. to yield the resultant lowering of 0.518 e.v. listed in Table I. The near-validity of the virial theorem is fortuitous here, as is evident from an inspection of the data pertaining to either the simple or the refined scheme separately. In order to compare the A M 0 and CI wave functions, we have expanded the former in terms of a set of orthonormal configuration functions given in the first column of Table 11. The expansion coefficients, in terme of Xo and AT*, are given in the second column of thie .table.l6 In the third and fourth
columns, numerical values of these coefficients are given in terms of the optimum A’s with the simple and refined A M 0 method, respectively. In the last column we have listed the nine coefficients of the CI wave function obtained by Parr, Craig, and Ross.’~ As a measure of the “resemblance” of the A M 0 and CI wave functions, Itoh and Yoshizumi4considered the “inner product integral”
With the simple and refined A M 0 function, we readily compute 9 = 0.983 and 9 = 0.986, respectively. Since these two wave functions are markedly different, as an inspection of the third and fourth columns of Table I1 shows, the quantity 9 can hardly be considered very revealing. We note that, while both A M 0 treatments exclude 44 and &, the coefficients of 410 and d14,in particular those of the refined A M 0 wave function, are of the same order of magnitude as those of d4,(P5, and 46 in the CI wave function. Here, as elsewhere, it is evident that it is difficult to decide a priori what configurations to include and which to exclude from a conventional CI treatment. This difficulty increases of course with the size of our system, whence the A M 0 method will become increasingly attractive. We plan to investigate such systems within the framework of the refined method described in this paper. Recently, Pauncz has computed the lowest triplet state in benzene using the simple A M 0 method.” We hope to obtain the same, using the refined method. This will not only allow for a further comparison between Ah10 methods and the CI treatment, but also will give us a first direct check against experimental data. Research to obtain singlet excited states also is in progress. Acknowledgment.-The author is indebted to Professor Per-Olov Lowdin, the originator of the A M 0 method, for his encouragement in the course of this investigation. This work has been supported in part by a grant from the Council on Research and Creative Work a t the University of Colorado. The computations were carried out on the CDC 1604 computer a t the Xational Bureau of Standards, Boulder, Colorado, with financial support of the Numerical Analysis Center of the Applied Mathematics Department a t the University of Colorado. The assistance of Mr. Hans M. Roder in programming the problem is gratefully acknomledged.
(15) As communicated to Itoh and Yoshirumi: yee ref 4 (16) The first nine’ of these coefficients already were obtained b) Itoh and Yoshieumi in the limiting case of the simple method, see
ref. 4. By substituting cos 28 for both XO and Xi?, it is readily verified that these two sets of coefficients become identical (17) R. Paunce, to be published
g
f q*abro\kcIdr
(25)
