1860
F. A. MATSEN AND D. J. KLEIN
that some of the computations were supported by research grants administered by Professor A. Kuppermann and Professor J. C. Polanyi, and by the University of Minnesota. We acknowledge helpful discussions with Professors Polanyi and Kuppermann, Professor L. R. Martin, Professor L. M. Raff, Professor E. S. Lewis, and
Professor E(.B. Wiberg. The 3-D perspective plotting routines were adopted from D. L. Nelson, Technical Report 553, Department of Physics, University of Maryland, 1966. The authors are grateful to Dr. David C. Cartwright for supplying them with his version of these programs.
Spin-Free Quantum Chemistry. IX.’ The Aggregate
Downloaded by UNIV OF SUSSEX on August 31, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/j100681a016
Theory of Polyelectronic Systems by F. A. Matsen* and D. J. Klein Molecular Physics Group, The University of Texas at Austin, Austin, Texas 78718
(Received November SO, 197‘0)
Publication costs assisted by the Robert A . Welch Foundation and the National Science Foundation
The aggregate theory of polyelectronic systems is applicable to interactions between shells, between atoms, and between molecules. The aggregate theory of an N-electron system consists of the following steps: (1) the construction of a zero-order Hamiltonian by partitioning N electrons among “aggregates”; (2) the construction of a vector space for the full Hamiltonian by induction on the zero-order spaces by S N ,the symmetric group on N objects; (3) the construction of kets which are symmetry adapted to SN; (4) the application of the Wigner-Eckart theorem for SN; and (5) a reduction in the number of primary matrix elements by the double coset decomposition of S N with respect to the group of the zero-order Hamiltonian. The double coset generators are permutations of electrons among aggregates which permit ready identification of exchange types.
1. Introduction The aggregate theory of polyelectronic systems is applicable to the study of the interaction between shells, between atoms, and between molecules. We present the formal theory in section 3. We partition the electrons of an N-electron system into sets called aggregates. The partitioning is carried out in such a way that the sum of the Hamiltonians for noninteracting aggregates provides a good zero-order Hamiltonian for the total system. We denote the group of the zero-order Hamiltonian by So. We take for the vector space of the full Hamiltonian the spaces induced on the zero-order eigenvectors by the symmetric group SN. The basis vectors in this space are then symmetry adapted to the sequence So E SN. The WignerEckart theorem reduces the problem to a sum over N ! primary matrix elements each of which involves a distinct permutation. In section 4 we make a double coset decomposition of 8, with respect to Xo and reduce the primary matrix elements to those which involve only a minimal number of interaggregate permutations. In section 5 we discuss approximations to the full aggregate theory and their areas of applicability. In certain c+ses the zero-order quantum numbers provide T h e Journal of Physical Chemistry, Vol. 7’6,h’o. 12, 1971
good approximate quantum numbers. The aggregate theory is the parent theory to a number of well known theories of polyelectronic systems. These theories and their relation to aggregate theory are discussed in section 6. The aggregate theory is developed in the spinfree formulation of quantum chemistry2 which is outlined briefly in section 2.
2. The Spin-Free Formulation The coarse structure of an N-electron system is accurately predicted with a spin-free Hamiltonian N
HSF=
N
CH, + i=
c 0 V(r0f;[AI)
P
(r Oro1 HsFP1 Y ‘Orro)
10TO)
4. Double Coset Decomposition The double coset decomposition of SN with respect to Sois SN =
@cq
(4.1)
4
where
symmetry adaptation is a set which is sequenceadapted3t o the chain
(3.13)
S N
A sequence-adapted matric basis element is denoted
e [XI ( ~ [ ~ ] O T O ) ( ~ ’ [ X ’ ] O ~ ’ O )
(p
where ro and r‘O are determined by the zero-order ket.
A convenient choice of matric basis elements for the
So
[V
(3.19)
(3.12)
[XI
[P-I ]
=
C,
=
SOGqSo
(4.2)
is the qth double coset and G , is a generator for that double coset. G, is an interaggregate permutation (except for the identity), so permutations over electrons can be replaced by the permutations of aggregate symbols (see Appendix 1). A sequence-adapted matric basis element can be expressed as a sum of products of double coset generators and the matric basis elements of So. Thus
where [A]’
[A,]
8
[AB]
8 [A,]
8 ..*
and = 1 to f[xlo;[xl
Here J[xlO;[xl is the number of times
of S N occurs
The Jozirnal o,f Physical Chemistry, Val. 76, N o . 12, 1971
(3) D. J. Klein, C. H. Carlisle, and F. A. Matsen, Advan. Quantum Chem., 5, 219 (1970).
SPIN-FREE
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QUANTUM CHEMISTRY
where I , is equal to NO divided by d,, the order of the intersection
G , S O G , - ~n SO
(4.4)
and
elxlo,oLo = e[xAJratA8
e[hB$BtB@
...
(4.5)
Downloaded by UNIV OF SUSSEX on August 31, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/j100681a016
Substituting the decomposition (4.3) into (3.18) yields
By the double-coset decomposition of SN with respect to So the sum over N ! permutations in (3.19) has been reduced t o a sum over interaggregate permutations. These terms are classified as follows: G, = ( A B ) , (AC), . . . single exchange and G, = (ABC) = ( A B ) ( B C ) ,( A B ) 2 , . . . nzultiple exchange. For a system of two aggregates we can choose the double coset decomposition as follows
G,
= (qNA
+ Q ) G , - q~ ~= 1 to min { N A , N B ) (4.7)
with
and (4.9)
5. Approximations Even though aggregate theory simplifies the general polyelectronic theory, it is often necessary to make approximations to render computation more tractable. Two types of approximations are frequently employed. A . T h e Neglect os Configuration Interaction. The interaction between different configuration vector is rigorously zero spaces ~ ( 7 0+), 70 = { v o KO;[X]O} only in the limit of noninteracting aggregates, for example, a system of widely separated aggregates. Configuration interaction can be neglected to a good approximation in those cases for which the zero-order energies are widely separated. Configuration interaction can also be neglected t o a good approximation between states of different [XI0 in those cases for which the aggregates are highly “localized,” i.e., there exists essentially zero differential overlap between the aggregates. An example of this case is provided by the derivation of the Heisenberg spin Hamiltonian as described in the following paper. B. T h e Neglect of Multiple Interaggregate Exchange Integrals. In this approximation one retains in eq 4.6 only the identity and the transpositions among the double-coset multipliers. The so-called multiple interaggregate exchange integrals are neglected. An example of this case is the computation of the London dispersion energies between aggregates at large interaggregate separation. Some estimates of the contributions
made to the several interaggregate exchange integrals can be obtained by considering their exponential behavior in the separated aggregate limit. For suitably accurate wave functions Herring4 has already considered this problem and found the exponential asymptotic behavior as given in Table I. I n this table algebraic and angle dependent factors for the asymptotic behavior of the integrals are neglected; Q ~ I Jis an orbital exponent and RIJ is the distance between the aggregates I and J. It is seen that the double coset multiplier terms interchanging the fewest indices among aggregates are most important in terms of their exponential asymptotic behavior. Table I : Double Coset Multipliers Interchanging Four and Fewer Indices” Number of P
G,
0
g
indices interchanged
Exponential asymptotic behavior
0
(IJ) (ij) (ZJK) (ijk) (ZJ)(ZJ) ( i j )(i’j’) ( I J ) ( Z K ) (ij)(i’k) (ZJ)(KL) (ij)(kl) (ijkl) (IJKL) All
2 3 4
~~IJRIJ+~’IJRIJ
4 4
eaIJRIJ+a’IJRIJ+~IKRIK+a’IKRIK
eaIJRIJ+aJKRJX+aKLRKL+aLIRLI
4
eaIJRIJ
eaIJRIJ+aJKRJIlc>
where
KO
=
{ICa, kb, . .
.I
and
so=Sl@S~@S~@ .
.
I
(4) C. Herring, Magnetism, 2b, 1 (1966). ( 5 ) See, for example, R. McWeeny, Proc. Roy. Soc. Ser. A . , 253, 242 (1959); M.Klessinger and R. McWeeny, J . Chem. Phys., 42, 3343 (1965); J. M.Parks and R. G. Parr, ibid., 28, 335 (1968). T h e Journal of Physical Chemistry, Vol. 76, No. I.%?, 1971
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F. A. MATSENAND D. J. KLEIN
The double coset generators are the group elements. One orbital choice is obtained by the optimally projected self-consistent field theory recently described by Gallup,elL Poshusta and I
