1878
F. A. MATSENAND B. R. JUNKER
Spin-Free Quantum Chemistry. XI.1 Perturbation Theory for Interaction Energies by F. A. Matsen" and B. R. Junker
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Molecular Physics Group, The University of Texas at Austin, Austin, Texas 78712 (Received November 80, 1070) Publication costs assisted by The Robert A. Welch Foundation and the National Science Foundation
Perturbation theory provides a method for a calculation of interaction energies between aggregates (atoms and/or molecules) which, in principle, does not require taking small differences between large quantities. I n such a theory consideration must be given to the permutational symmetry. The spin-free formulation of the polyelectronic problem employs directly the permutational symmetry of the problem and is therefore well suited for the treatment of interaction energies. Special restrictions on the perturbation function lead to the Hirschfelder-Van der Avoird, Hirschfelder-Silbey, Murrell-Shaw, and Amos-Musher formalisms. The perturbation energies in these formalisms are a function of those restrictions and only in infinite order is the dependency removed.
1. Introduction In principle, accurate eigenvalues and eigenfunctions
nuclei system commutes with the (symmetric) group SN of permutations on the electron indices, i.e.
of an N-electron spin-free Hamiltonian can be obtained by diagonalizing the representation of the Hamiltonian in a vector space spanned by a large basis set. In practice this program has been carried out only for systems with small numbers of electrons and nuclei. As the number of electrons and/or nuclei increases the tractability and the accuracy of a calculation decrease rapidly. Many important properties of polyelectronic systems are determined solely by the interaction energy between aggregates (atoms and/or molecules). From total energy calculations the interaction energies are obtained by subtracting two large numbers. Since interaction energies are small, perturbation theory offers in principle the possibility of computing the interaction energy directly.za That is, the problem may admit formulation in such a may that the zero-order energy is the energy of the noninteracting aggregates and the sum of the higher order perturbation energies is the interaction energy. It is hoped that only a few terms are required in the perturbation expansion. I n this problem proper attention must be given to the permutational symmetry. In section 2 we present in the spin-free formulationzbthose aspects of permutational symmetry which are important for the perturbation problem. In section 3 we develop a general perturbation theory in the spin-free formulation for systems of aggregates. There occur difficulties in the general perturbation theory which can be alleviated by placing restrictions on the perturbation wave function. This general restricted perturbation theory is discussed in section 4.
2. Spin-Free Quantum Chemistry The spin-free Hamiltonian, R, for an N electron, M The Journal of Physical Chemistry, Vol. 76, N o . 1.9, 1972
LHJPI
= OJ
E
SN
(2.1)
As a conskquence of (2.1) and the Wigner-Eckart theorem, the eigenfunctions to H are symmetry adapted to SN and the Schroedinger equation has the form
Hlx [Air)
=
[']Ex/X[A]r)
(2.2) Here, [A], called the permutation quantum number, identifies the irreducible representations of SN, r is the permutation degeneracy index, and x represents all other quantum numbers. Since the set { / X [ Y ] ~ r) , = 1, . , , f[']] is a degenerate set of eigenfunctions of H, we suppress r. The permutation states which occur in nature are those for which
(2.3) and are called Pauli-allowed permutation states. The remaining states are called Pauli-excluded permutation states. An allowed state in the conventional spin formulation is labeled by the spin quantum number S which is related to a Pauli-allowed permutation quantum number by s = N- - p [A] =
[2pJlN--2p]
2
The multiplicity is then given by m=N-2p+l
(2.4)
(2.3
(1) Supported by the Robert A. Welch Foundation of Houston, Texas, and the National Science Foundation. (2) See J. 0.Hirschfelder, Chem. Phys. Lett., 1, 325 (1967),for a list of references in the literature concerning this problem; (b) F. A . Matsen, Adsan. Quant. Chem., 1, 69 (1964); J . Amer. Chem. Soc., 92, 3525 (1970); D.J. Klein, Int. J. Quant. Chem. Symp., 4, 271 (1971).
SPIN-FREE QUANTUMCHEMISTRY
1879
Both Pauli-allowed and Pauli-excluded spin-free eigenfunctions of H exist. We consider now a system of aggregates (atoms and/or molecules) labeled A , B , . . . with N a , NB, . . . electrons, respectively. At infinite aggregate separation the energy for this system can be computed from a Hamiltonian
. . . NA) +
H(O) = HA(^, 2,
HB(NA
+ 1, . . . N A + NB) + . . .
(2.6)
The eigenvectors are products of eigenvectors of HA,
be taken as a perturbation Hamiltonian, El('). As a consequence of (2.11) any perturbation formalism is nonunique since the left-hand side of (2.11) is zero order and the right hand side is first order. Thus (2.11) could be used to define different perturbation schemes by shifting terms between the various orders. Equation 2.12 is the Schroedinger equation with the separated aggregate energy explicitly included. I n order to express the dependence of Ix[h])on the separated aggregate state (yC0)) we define a perturbation series for Ix[A])
HB, . . .
m
ly(O))
IxA[XA]~A)lXB[XBlrB).
*
1
(2.7)
and the eigenvalues are sums of the eigenvalues for Downloaded by TEXAS A&M INTL UNIV on August 31, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/j100681a019
HA,
HB,
' '
.
ex (O) = ['AIExA+ ['BlExB + . , ,
(2* 8)
I n (2.6) we have made a specific assignment of the electrons to the various nuclei. This is, of course, not unique. For finite interaggregate separations we define the interaction energy of the (X,[h])th state as
A[']Ex E [']EX-
(2.9)
where EX(') is the energy of the separated aggregate state with which the (X,[h])th molecular state correlates. We define the interaction operator for H and H(O) as "(1)
=
H - H(0)
(2.10)
We note that while H commutes with the elements of S N , H(O) commutes only with the elements of S(O) = S A 8 S B 8 . . ., a subgroup of S N . Here Sa is the symmetric group for aggregate A , SBfor aggregate B, . . . . As a consequence
[H'O',P]
=
[P,H'l'] # 0
(2.11)
for all P s(O). An equation equivalent to the Schroedinger eq 2.2 is
[(H'O) -
+ H(l) ('']Ex -
EX(O))]IX[X]) = 0
(2.12)
We formally compute the interaction energy by applying (y(O)I to (2.12). Then since (y(O)/(H(O)-
EX'O))~X[X])
= 0
IX[X])
=
e['] aC IT(')) =O
e[']ly)
(3.1)
where IT(')) is the separated aggregate configuration with which the (X [A])th molecular state correlates and e''] is the projector for the [A]th permutation state. If the energy, [']Ex, is also expanded in a perturbation series m
=
['I&
['lEX(Z)
(3.2)
1.=0
and if (3.1) and (3.2) substituted are substituted into (2.12) we obtain
e[']
m
+ H(l) -
[(HO -
i=O m
[XIEX(3)
+ €x(o)]/y(2)) = 0
(3.3)
3=0
Note that since Jy(O))is the wave function of the separated aggregate state with which the (X[A])thmolecular state correlates ['I
EX
(0)
= ,x(o)
(3.4)
and m
= b1Ex - ,x(o) =
['IEx(%) i=l
(3.5)
With the requirement above the higher order equations are
e['l[(H(O) - ,x (0)) I Y ( l ) ) (H(') - [ ' ] ~ x ( ~ ) ) I y ( = ~ ) 0) ] (3.6a)
+
ebl[(H(o)- ,x (0)>Ir'"')
+
(2.13)
we have by (2.12)
3. General Projected Perturbation Theory for Interaction Energies Equation 2.10 defines a nonunique partitioning of the total Hamiltonian into a term which may be chosen as the zero-order Hamiltonian, H(O),and a term which may
Note that these equations do not differentiate among different states of the same symmetry [A] which arise from the same separate aggregate state Consequently, even if more than one state for a given [ A ] does correlate with y(O) there can be obtained one and only one energy corresponding to a state with symmetry [A]. A major difficulty with eq 3.6 is that the nth order wave function is required to determine the nth order The Journal of Physical Chemistry,Vol. 76,No.18,1971
F. A. MATSENAND B. R. JUNKER
1880 energy. This differs from normal Rayleigh-Schroedinger2&perturbation theory where the nth-order wave 1). function determines the energy to order (2n This has occurred since we have introduced the symmetry explicitly by means of e[’] and thus one cannot form the bracket of (3.6) with (y(O)I and eliminate (H“’ - e ~ ( ’ ) ) / y ( ~ ) )We . call this the symmetry problem. A possible solution to this difficulty is to substitute from (2.11)
+
e[XIH(0)= H(0)e[A1+ H(l)e[Al- e[AIH(l) (3.7) into (3.3) for i 2 1. Then the perturbation equations ebI(H(0)Downloaded by TEXAS A&M INTL UNIV on August 31, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/j100681a019
(“(0)
= 0
E x(o))Iy(o))
(H‘” = e ~ ( ’ ) ) l y ( ~ = )) l ~ ( O ) { h ] ) - Ex(o))Iy(l)) + e[XI(H(1)[XI EX(1)
(3.8~1)
- Ex(0))e[Xl(y(l)) + -
Substitution of (4.1) into (4.3) yields (2.14). The strategy now is to expand AFXIEx, IT), and I r { X ) ) in perturbation series like (3.1) and (3.2). This leads to the following perturbation equations.
(“0)
become
(“0)
The interaction energy is seen to be given by
(“0)
- eX)JY(n))
IY
(0))
= IT(’){X})
(4.4a) (4.4b)
>-
+ e[XIH(l)Jy(n-l)
erX](H(l)- [X]ax(l))l~(o)) = 0 (3.8b) eX(0))e[hlly(fl)) + H(l)e[Ally(n-l))n
lXlex(d m=l
(A)
e IY
(n-4)
= 0
(3.84
A remark similar to that following (3.6) applies here. We obtain from (3.8b) [XIEX(1)
=
(y(0)(e[hlH(l) I ) (0) /(Y
and from (3.8~) [XIEX(n) = (( ( 0 )/H(l)e [XI I n-1
(n- 1)
(0)
le bl IY(0)
(3.9)
[XIex(0)
>-
le
b I e X (d(Y (0) [All,, (n -m)))/
m=1
(y(o)le[A1ly(o)), n > 1 (3.10) Assuming (3.8b) and ( 3 . 8 ~ have ) nontrivial solutions, they provide a means for determining Iy(l)) and respectively. The equation
is the formal first-order solution to eq 3.8.1.
4. General Restricted Perturbation Theory In this section we show a second method by which the symmetry problem discussed in section 3 can be resolved by placing ti very general restriction on the perturbation function. We call this the general restricted perturbation theory. I n this development we follow closely a development outlined by Hirschfelder.2a The general restriction which we impose on the perturbation function 17) is
It is clear that l y { X ) ) does not, contain If eq 4.1 is added t o (3.3) we have The Journal
of
Note that again only one state of a given symmetry [ X I can be obtained, even if more than one state of [ X I symmetry correlates with a given separated atom state ly(O)). Again if 1 ~ ‘ ~ ) is ) taken to be the separated aggregate state with which the (X[X])th molecular state correlates and if ~ Y ( ~ ) ( X } is ) defined as the zero vector
[ A ] symmetry.
Physical Chemistry, Vol. 76, No. I $ , 1971
= ,x(o)
(4.5)
If the bracket of ( 4 . 4 ~is ) formed with (y(O)/,the following perturbation energies are obtained. [XIEX(n) = [(Y (0)le [XI €3 (1) IY(n-1)) n-1
IXIEx(m)( y (0)le [XI l y j ( n - m ) -
m-1
(Y (0)IY(n){ A ) )l(Y(o)le[X1lr‘o’)(4.6) The significance of the last term in (4.6) is that it corrects for the erroneous symmetry which was added in by the left-hand side of (4.1). There exist an infinite number of restrictions of the type of (4.1) which could be imposed on the perturbation wave function. Lekkerkerker and Laidlaw, and Kleinzb discuss the particular restrictions which lead to the Hirschfelder-Silbey5p6(HS), hfurrellShaw,’ R’Iusher-Amos,s and Amos-Musherg perturbation formalisms. The “optimum” or “correct” choice of the restriction (4.1) is certainly not obvious, if indeed one does exist. The suggestion which has been made (3) H . N. W. Lekkerkerker and W. G. Laidlaw, J. Chem. Phys., 52, 2953 (1970). (4) A. T. Amos, Chew. Phys. Lett., 5, 587 (1970). (5) J. 0. Hirschfelder, ibid., 1, 363 (1967). (6) J. 0. Hirschfelder and R. Silbey, J. Chem. Phys., 45, 2188 (1966). (7) J . N. Murre11 and G. Shaw, ibid., 46, 1768 (1967). (8) J. I. Musher and A. T. Amos, Phys. Rev., 164, 31 (1967). (9) A. T. Amos and J. I. Musher, Chem. Phys. Lett., 3 , 721 (1969).
DIFFRACTION O F
LIGHTBY
that calculations might resolve this problem is not very practical because of the infinite number of possibilities.
5. Summary We have presented a spin-free formulation of perturbation theory for interaction energies. We discussed two techniques for treating the symmetry problem raised in section 3. One method utilized explicitly the nonuniqueness resulting from (2.11) while the
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1881
NONAQUEOUS ORDERED SUSPENSIONS
1~‘~’)
second method placed certain restrictions on the for n 2 1. The fact that there exist an infinite number of possible perturbation schemes and that there exist no criteria outside of computation for evaluating them, poses a serious problem in perturbation theory for interaction energies.
Acknowledgments. The authors acknowledge helpful discussions and hospitality at the Theoretical Chemistry Institute at the University of Wisconsin.
Diffraction of Light by Nonaqueous Ordered Suspensions
by P. A. Hiltner,* Y. S. Papir, and I. M. Krieger Department of chemistry and the Division of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44108 (Received November 1% 19’70) Publication costs assisted by the Public Health Service
A technique is described for resuspending latex particles in nonaqueous liquids. Suspensions of a monodisperse latex in some polar liquids are iridescent and give Bragg diffraction peaks. The particle separation D in the ordered array, the particle diameter DO,and the volume fraction @ obey the relationship ~ ( D / D o ) ~ = 0.74, the value 0.74 being the volume fraction occupied by spheres in a close-packed arrangement. The order is attributed to electrostatic repulsion between particles as a result of partial dissociation of ionic surface groups. Intrinsic viscosity measurements indicate that swelling is negligible in most polar liquids. Suspensions in nonpolar liquids are either unstable or highly swollen and do not show Bragg diffraction.
Bragg diffraction by ordered colloidal suspensions was reported in an earlier paper.l llleasurements on electrolyte-free monodisperse latexes showed that the particles are in a close-packed arrangement which persists throughout the suspension, even when the particles are several diameters apart. The latex particles are charged, owing to bound initiator fragments, and the long-range order is attributed to interaction of the electrical double layers. I n the absence of shielding electrolyte, the electrostatic repulsion was found to be effective over distances of several particle diameters. The magnitude and range of the interparticle potential were varied experimentally by addition of electrolyteq2 The interparticle potential, as well as other properties of the suspension, should depend on the nature of the suspending medium. The present work describes the preparation of stable latex suspensions in nonaqueous liquids and their characterization by optical diffraction.
rene with 0.5-10.0% divinylbenzene was emulsionpolymerized in the presence of both ionic and nonionic surfactants; polymerization was initiated by the thermal decomposition of potassium persulfate. The resulting aqueous suspensions were about 50% polymer by volume and highly iridescent. Particle sizes obtained by electron microscopy ranged from 0.15 to 0.25 p ; the uniformity index (ratio of weight average diameter to number average diameter) was always less than 1.01. To redisperse the latex particles in nonaqueous media, the aqueous suspension was initially deionized by addition of a monobed ion-exchange resin (Amberlite AIB-3, Rohm and Haas Co.) in the ratio of 1 g of resin per 25 g of latex. After 24 hr the resin was removed by filtration. This procedure removes both
Experimental Section
(2) P. A. Hiltner and I. M. Krieger, “Order-Disorder Behavior in Monodisperse Colloids,” in “Polymer Colloids,” R. Fitch, Ed., Plenum Press, New York, N. Y . , in press. (3) Y . S. Papir, M . E. Woods, and I. M. Krieger, J . Paint Technol.,
Details of the preparation of monodisperse crosslinked latexes have been described p r e v i ~ u s l y . ~Sty-
(1) P. A. Hiltner and I. M. Krieger, J . Phys. Chem., 73, 2386 (1969).
42, 571 (1970).
The Journal of Physical Chemistry, Vol. 76, No. 18, 1972
