J. Phys. Chem. 1996, 100, 10595-10599
10595
Inverse Strategies for Molecular Design Christoph Kuhn and David N. Beratan*,† Department of Chemistry, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15260 ReceiVed: February 20, 1996X
An “inverse” molecular design strategy is described to assist in the development of new molecules with optimized properties. This approach is based on a molecular orbital view and can be used to tailor ground state or excited state properties subject to particular constraints. In this scheme, wave functions are sought that optimize a chemical or electronic property, and then a Hamiltonian is constructed that generates these optimized wave functions. Analysis of the chemical properties in the optimized systems may suggest new synthetic targets. Examples are presented that optimize the transition dipole moment in some simple structures.
I. Introduction A broad challenge in materials chemistry is to design structures that optimize a particular molecular property subject to practical constraints, such as thermal stability or optical gap. Recent attempts to build consistent and predictive quantumchemical models have led to useful structure-function relationships in many areas of materials chemistry.1 While successful, these relationships often succeed for limited families of related molecular structures. In developing structure-function relationships, the conventional molecular design strategy often involves minor variations on a known motif, computation of the properties under investigation, chemical synthesis of suggested targets, and measurement of the altered properties. We refer to this molecular design strategy as a “direct” approach.2 The goal of the present research is to address the challenge of mapping out classes of molecules that will optimize a given property in a more global sense. This goal is one of “inverse”3 rather than of direct molecular design. A considerable advantage of the inverse strategy is illustrated schematically in Figure 1. The direct molecular design strategy usually maximizes properties in a fairly narrow zone of “molecular space.” As discussed below, this direct strategy can indeed be carried out in a more global manner and could be developed with many of the advantages of inverse methods. Nevertheless, the goal of inverse strategies described here will be to sample more of the molecular space, searching for global property maxima in classes of structures (that might not have been considered with other methods) subject to convenient constraints. Most quantum chemical approaches to electronic materials begin with a known chemical structure and the properties of interest are calculated from the eigenstates and energy eigenvalues of the corresponding Hamiltonian. To optimize the property of interest, one is motivated by experiment at the outset and considers small variations on the molecular structure. If the property improves upon the proposed structural modification, synthesis and characterization of the target molecule are warranted. The optimum structure within the class of molecules might be found by repetition of this direct strategy (Figure 1, left). In contrast to the direct strategy, the inverse method asks the question what structure (e.g., size, connectivity, or atom types) characterized by its corresponding Hamiltonian optimizes a given property (e.g., linear and nonlinear optical properties,1a,b † X
E-mail:
[email protected]. Fax: (412) 624-8552. Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(96)00518-7 CCC: $12.00
Figure 1. Direct (left) and inverse (right) strategies for molecular design.
conductivity,1e magnetic properties,1f etc.). The theoretical problem is somewhat ill-posed without some specific constraints. For example, in molecules connectivity is local. This kind of constraint can be added in two ways, by limiting the nature of the Hamiltonian matrix itself or by constraining the bond order matrix. In the inverse strategy described here, the energy spectrum must be specified as well. We first consider properties that are sufficiently weakly dependent upon the energy spectrum and thus we do not carry out energy spectrum optimization. In the design of optical or electronic materials, it may be convenient to set the energy spectrum to fix bandwidths, band gaps, and the density of electronic states. To optimize a property, we tailor the LCAO-MOs (linear combination of atomic orbitalsmolecular orbital) wave function coefficients (subject to connectivity constraints). We then construct the Hamiltonian that would generate these ideal wave functions for the prescribed energy spectrum. The Hamiltonian matrix elements are determined by the set of all wave function coefficients and the prescribed energy eigenvalues. The goal of this inverse strategy for molecular design is to suggest classes of structures that will optimize an observable without violating our sensibilities about valence and energetics in covalent compounds (Figure 1, right). II. Molecular Design Strategies In general, the problem of molecular design is one of nonlinear optimization.4 Changes in molecular structure, and thus changes in the matrix elements of the corresponding Hamiltonian, lead to changes in the wave functions, energy eigenvalues, and derived properties. This relationship between quantities is certainly nonlinear. The aim of this paper is to develop techniques to suggest the optimum class of molecules for a particular property. This challenge is one in which linear relations, valid in restricted regimes, are not useful. (a) The “Direct” Method of Property Optimization. A direct strategy for molecular design begins with a chemical structure (e.g., a chain of N carbon atoms) and, in an © 1996 American Chemical Society
10596 J. Phys. Chem., Vol. 100, No. 25, 1996
Kuhn and Beratan 0). The Hamiltonian matrix may be written
(
0 -1 0 H ) -1 R2 β2 0 β2 R3
)
(3) (2) (3) The problem is to optimize O23(R2,R3,β2) ) c(2) 2 c2 + 2c3 c3 by varying the Hamiltonian elements. Figure 2 shows the hypersurface of the transition dipole moment O23(R2,R3) for β2 ) -1. For R2 > -5.0 the optimum O23 ) -0.967:
Figure 2. Example 1. Linear chain, 3 sites, 3 LCAOs: the surface of the transition dipole moments O23(R2,R3) for β2 ) -1.
independent electron (or mean field) calculation, one constructs the Hamiltonian or Fock matrix associated with that structure.5 Diagonalizing this matrix generates N energy eigenvalues k with the corresponding LCAO wave functions ψk ) ∑Ni c(k) i φi. The atomic orbitals φi are located at positions ri ) axi along the x-axis where a is the unit of length. The orthonormality (m) condition ∑Ni c(k) ) δkm is fulfilled since the matrix is real i ci and symmetric. To summarize the direct method,
{
}
molecule f H )
R1 ) 0 β1 ) -1 γ1 ... ... R2 β2 β1 ) -1 ... ... γ1 β2 ... ... γN-2 9 8 diagonalize ... ... ... ... βN-1 γN-2 βN-1 RN ... ...
(
) {{ } (
{c(k)} ) R‚PR
{{k}, {ci(k)}} (1)
For simplicity, the Hamiltonian H can be thought of as a simple Hu¨ckel Hamiltonian. Our freedom to set the energy zero and the energy units allows us to set R1 ) 0 and β1 ) -1. We now consider an observable represented by the operator O ˆ . The property or matrix element of the operator O ˆ ({k}, (c(k) ), {r }) can be evaluated using these eigenstates and energy i i eigenvalues. For example if O ˆ represents the electric dipole operator, its matrix elements (neglecting overlap corrections) are N
Okm ) ea∑xici(k)ci(m)
(2)
i
where e is the electron charge. In the following discussion we set ea ) 1. Scanning through all the possible values of the entries in the Hamiltonian matrix, one hopes to find the global extremum of the O hypersurface. The Hamiltonian has Et ) (N2 + N)/2 - 2 entries, although many of the non-nearestneighbor elements are expected to be small. In the case of a linear chain, the number of non-nearest-neighbor interactions is Cu ) {(N - 1)(N - 2)}/2; there are only Eu ) Et - Cu ) 2N - 3 degrees of freedom in the Hamiltonian that includes nearestneighbor interaction between sites. Direct strategies that scan all or limited ranges of Hamiltonian matrix elements have been used recently in nonlinear optics.1a,6 As the number of sites grows, the complexity of identifying minima grows also, and the problem becomes one of finding minima on a convoluted surface7 and may require statistical sampling techniques.4,8 Example 1. We apply the direct method to the problem of a linear chain of three orbitals at positions {xi} ) {0,1,2} and neglect non-nearest-neighbor interaction (H13 ) H31 ) γ1 )
)}
0 -1 0 -5.4 0.18 0.97 0.18 0 , 0.71 0 -0.71 H ) -1 -5 -1 f 0 -1 0 0.37 0.68 -0.26 0.68 Note that the energy spectrum (column vector above) at every point on this surface is different. Here the global property surface can be computed directly and the property maximum readily identified from the surface. (b) The “Inverse” Method. The inverse method of molecular design optimizes the property of interest by varying the wave function coefficients. The energy spectrum is held fixed. The LCAO wave function coefficients for each MO are built up from an orthonormal set of wave functions. Optimizing this set of wave function coefficients poses the same challenge as optimizing the orientation of a vector in N-dimensional space.9 As such, we can describe the optimization as constructing the rotation matrix R that rotates the initial set of atomic orbitals into a set of LCAO wave functions {ψk} ) R‚PR{φi}. At the end of the procedure the optimal eigenvectors (eq 3) are obtained. (3)
PR is a permutation operator depending on R explained below and
R ) R(N-1),N(ϑ(N-1))‚R(N-2),(N-1)(ξ(N-1)(N-2)/2)‚......‚ R2,(N-1)(ξN-1)‚R2,N(ϑ2)‚R1,2(ξN-2)‚...‚R1,(N-1)(ξ1)‚R1,N(ϑ1) (4) is the rotation constructed by factoring into Mχ ) [N(N - 1)]/2 in-plane rotations.10 In effect, performing these rotations allows us to sample, in a systematic manner, the entire molecular wave function space spanned by all possible weightings of pairs of atomic orbitals in the molecular wave function.
(
1p-1 0 0 0 0 -sin(χ) 0 0 cos(χ) 0 1q-p-1 0 Rp,q(χ) ) 0 0 0 cos(χ) 0 sin(χ) 0 0 1N-q 0 0 0 0
)
(5)
Here 1 e p < q e N and 1k is a k-dimensional unit matrix. 0 is a corresponding rectangular zero matrix. This in-plane rotation is defined by the general angle χ within the plane spanned by the two unit vectors I(p) and I(q). Examining terms in the product (4) with q ) N, there are Mϑ ) N - 1 different rotations with angles χ ) ϑp in the range -π < ϑ < π (longitude angles). For terms with q * N, there are Mξ ) {(N - 1)(N 2)}/2 different rotations with the corresponding angles χ ) ξpN-q-(p-1)(p+2)/2 in the range of -π/2 < ξ < π/2 (latitude angles). Note that Mχ ) Mϑ + Mξ. The prescribed M ) N 2 energy eigenvalues, {k} ) {1 ) -1, 2 ) 0, 3 > 0, ..., i < i+1, ..., N}, are in order of increasing energy. Our freedom to set the energy zero and the energy units allows us to set 2 ) 0 and 1 ) -1.
Inverse Strategies for Molecular Design
J. Phys. Chem., Vol. 100, No. 25, 1996 10597
The correct nodal structure of the wave functions {c(k)} according to the corresponding prescribed energy eigenvalues {k} is guaranteed by the permutation operator PR.11 Both the permutation and rotation operators leave the orthonormality of the wave functions invariant. Given this set of LCAO wave functions {c(k)} from eq 3, the Hamiltonian H is obtained from N
Hij ) ∑ci(k)kcj(k)
(6)
k
Molecular connectivity is local, so the Hamiltonian elements connecting distant sites will be set to zero. Let us assume that (k) we have Cu connectivity constraints Hij ) ∑Nk c(k) i kcj ) 0 for specific atomic orbitals i and j. With M prescribed energy eigenvalues fixed as well, the highly structure-constrained hypersurface of the property O is explored: scanning through the Mχ angles the global extrema subject to Cu connectivity constraints is found numerically by either the Lagrangian multiplier method12 or, alternatively, by finding roots that satisfy the connectivity constraints first and then optimizing subsequently (in the example below, there are three angles and one connectivity constraint; the constraint fixes one angle). Applied to larger systems (N g 4), optimization subject to the constraints (Cu g 2) can be tedious, and the latter method may even become impossible to solve in general.4 Statistical techniques as mentioned in section IIa may be needed to address this optimization problem. In the case of a linear chain that excludes Cu ) {(N - 1)(N - 2)}/2 non-nearest-neighbor interactions, the resulting Hamiltonian is defined by Mu ) Mχ - Cu + M ) 2N - 3 free parameters, the same number of degrees of freedom as in the direct method. Example 2. We apply the inverse method to the linear chain of three orbitals at positions {xi} ) {0,1,2} excluding nonnearest-neighbor interaction. The given energy eigenvalues {k} ) {-1,0,3 > 0} are in order of increasing energy and the orthonormal space of LCAO coefficients is {c(k)} ) R(ϑ1,ϑ2,ξ)‚Pϑ1,ν2,ξ parametrized by
R(ϑ1,ϑ2,ξ) ) R2,3(ϑ2)‚R1,2(ξ)‚R1,3(ϑ1) )
(
)(
)
1 0 0 cos(ξ) -sin(ξ) 0 0 cos(ϑ2) -sin(ϑ2) sin(ξ) cos(ξ) 0 × 0 sin(ϑ2) cos(ϑ2) 0 0 1
(
cos(ϑ1) 0 -sin(ϑ1) 0 1 0 sin(ϑ1) 0 cos(ϑ1)
)
There are 3‚2 ) 6 permutations P(ϑ1,ϑ2,ξ) of unit vectors
( )
1 0 0 I123 ) 0 1 0 0 0 1 Two permutations
{ } { } { }
I I I P(ϑ1,ϑ2,ξ)(I123) ) I123 or I312 or I231 321 213 132
depending on (ϑ1,ϑ2,ξ) give the correct nodal structure for the wave functions according to the sequential order of the corresponding energy eigenvalues. The connectivity constraint (1) (3) (3) H13(ϑ1,ϑ2,ξ(p)) ) - c(1) 1 c3 + c1 3c3 ) 0 is fulfilled for specific values ξ(p)(ϑ1,ϑ2). The problem is to optimize O23(ϑ1,ϑ2) (3) (2) (3) ) c(2) 2 c2 + 2c3 c3 . Figure 3 shows the hypersurface of
Figure 3. Example 2. Linear chain, 3 sites, 3 LCAOs: the transition dipole moment surface O23(ϑ1,ϑ2) for 3 ) 1.
transition dipole moments O23(ϑ1,ϑ2) for the value 3 ) 1. The minima are at
ϑ1 )
{
0.62 , ϑ ) 0.62, ξ ) 0.52 0.95 2
with
O23 ) -0.71,
{{ } (
-1 0.50 0.71 0.50 0 , 0.71 -0.01 -0.71 1 0.50 -0.71 0.50
(
)}
f
-0.01 -0.71 0 H ) -0.71 0 -0.71 0 -0.71 0.01
)
Note that every point on this surface O23(ϑ1,ϑ2) has the same fixed spectrum of energy eigenvalues and satisfies the connectivity constraint. Thus, viewing the property surface allows us to optimize a molecular property subject to the constraints. (c) The “Hybrid” Method. The direct and inverse methods by themselves are similar in complexity. Both have the same number of degrees of freedom (i.e., matrix elements or wave function coefficients, respectively) for searching the optimum on the hypersurface of the desired property. The hybrid method combines aspects of both the direct and inverse methods (Figure 4). The property itself is the free variable that drives the optimization: fixing its value, the matrix elements and wave function coefficients are obtained by an iteration starting from an initial guess and leading to self-consistency. An optimum of the property is obtained by lowering the value of the property steadily toward self-consistency. The global optimum of the property in the case of a multiminimum hypersurface (e.g., example 4) can, in principle, be found by enforcing additional constraints to simplify the property surface to one minimum13 followed by subsequent relaxation of these constraints (Figure 5). Figure 4 shows the flow chart of the hybrid method in detail. With the diagonalization of a guess Hamiltonian, iteration is initiated until convergence to self-consistency of (m) ) (0) the wave function orthonormality constraint ∑Ni c(k) i ci δkm; (i) the given value of the property Ogiven; (ii) the given energy eigenvalues {(k)given}; (iii) the given connectivity constraints H{(i,j)given} ) 0. Diagonalizing a real-symmetric matrix, the wave functions obtained are orthonormal (condition (0)). Fixing the given value of the property (condition (i)), an appropriately chosen wave function coefficient is changed (in the case of the transition (m) by its inverse dipole moment Okm ) ∑Ni xic(k) i ci
10598 J. Phys. Chem., Vol. 100, No. 25, 1996
Kuhn and Beratan
Figure 6. Example 3. Linear chain, 4 sites, 4 LCAOs. Matrix elements H{ii, 12, 23, 34} as a function of the transition dipole moment O23. The guess Hamiltonian gives O23 ) -0.68 but not the prescribed eigenvalues (thus it is not self-consistent). The value of O23 is continuously lowered while iterating to self-consistency until the absolute minimum O23 ) -1 is reached.
Figure 4. Hybrid method dovetailing the direct and inverse methods (see flow chart). An interation is initiated by diagonalization of a guess Hamiltonian continued until convergence to self-consistency of (0) an orthonormal set of wave functions, (i) the given value of the property, (ii) the given energy eigenvalues, (iii) fulfilling the connectivity constraints. Changing LCAO coefficients during iteration causes orthonormality to be lost, but the inverse method returns a realsymmetric matrix with an orthonormal set of wave functions.
constraints are enforced. The subsequent diagonalization returns an orthonormal set of wave functions. This procedure converges only if there is a self-consistent solution fulfilling all conditions (0) and (i)-(iii). The value of the property is then steadily lowered while maintaining self-consistency. Local minima (e.g., Figure 5, solid line minimum on the left) are escaped by appropriately chosen additional constraints (Figure 5, following the dashed line of constraints B) and their subsequent relaxation (Figure 5, solid line, minimum on the right). Example 3. We apply the hybrid method to the problem of a linear chain of 4 orbitals at positions {xi} ) {0,1,2,3}. Nonnearest-neighbor interactions are excluded so we have the connectivity constraints H{13,14,24} ) 0. We preset the eigenvalues {k} ) 1/2{-3,-1,1,3}. Figure 6 shows the matrix elements as a function of O23. The optimum value O23 ) -1 (3) (2) (3) (2) (3) of the property O23 ) c(2) 2 c2 + 2c3 c3 + 3c4 c4 is reached with
(
)
0 0 -x3/2 0 -1 0 -x3/2 H) f 0 -1 0 -x3/2 0 0 0 -x3/2 0
{}
(
-3 -1 1 , {1/2 1 2x2 3
x3 x3 1 x3 1 -1 -x3 x3 -1 -1 x3 1 -x3 x3 -1 1
)
Example 4. We apply the hybrid method to a square lattice of 3 × 3 orbitals at positions Figure 5. Outline of the hybrid method. Beginning with a guess Hamiltonian and a fixed value of the property O, self-consistency is obtained by iteration (Figure 4). The value of the property is then continuously lowered while iterating to self-consistency. Local minima are escaped by appropriately chosen additional constraints (dashed line, constraints B) and their subsequent relaxation (solid line, minimum on the right). N
(O ˜ km)given - ∑xici(k)ci(m) cj(k) )
i*j
xjcj(m)
see examples 3 and 4) and orthonormality is lost. Given the spectrum (condition (ii)) the inverse method returns a realsymmetric matrix and, with condition (iii), the given connectivity
{ }{
(0,2) (1,2) (2,2) 7 8 9 (x,y) 4 5 6 ) (0,1) (1,1) (2,1) 1 2 3 (0,0) (1,0) (2,0) connected as
{ }
}
7 - 8 - 9 | | | 4 - 5 - 6 | | | 1 - 2 - 3 by the connectivity constraint that Hij ) 0 if a bar does not join sites i and j above. We preset the eigenvalues {k} ) {-1,0,1,2,3,4,5,6,7} and optimize the property O56 ) ∑i(xi + (6) yi)c(5) i ci . The absolute minimum O56 ) -1.5185 is obtained (Figure 7) assuming additional connectivity constraints
Inverse Strategies for Molecular Design
J. Phys. Chem., Vol. 100, No. 25, 1996 10599 in areas where the intrinsic property is relatively well understood but key high-performance materials are not forthcoming. Acknowledgment. We thank Drs. I. V. Kurnikov, S. Priyadarshy, and S. M. Risser for helpful discussions. This work was supported by the National Science Foundation (Grant CHE9257093) and the Department of Energy (Grant DE-FG3694G010051). Additional assistance from the University of Pittsburgh Materials Center is gratefully acknowledged. References and Notes
Figure 7. Example 4. Wave function coefficients for the square lattice of 3 × 3 sites, 9 eigenvalues. Filled (empty) circles ) positive (negative) amplitude, star ) 0 amplitude. The global optimum of the transition dipole moment O56 ) ∑i(xi + yi)ci(5)ci(6) is obtained by assuming the addition connectivity constraint H{12, 23, 36, 45, 58, 69} ) 0 and subsequently relaxing it.
H{12, 23, 36, 45, 58, 69} ) 0 and subsequently relaxing these additional constraints.
(
Hmin ) 2.8 0 0 -1.5 0 0 0 0 0 0 3.0 0 0 -2.5 0 0 0 0 0 0 5.0 0 0 0 0 0 0 -1.5 0 0 2.3 0 0 -2.5 0 0 0 -2.5 0 0 3.4 -1.5 0 0 0 0 0 0 0 -1.5 1.5 0 0 0 0 0 0 -2.5 0 0 4.0 -2.6 0 0 0 0 0 0 0 -2.6 2.0 -1.5 0 0 0 0 0 0 0 -1.5 2.9
)
Relaxing the additional constraints changes the results slightly in this case. Assuming another set of connectivity constraints H{12, 25, 36, 47, 58, 89} ) 0 and their subsequent relaxation gives only a local minimum O56 ) -1.45611. III. Conclusions The general problem of molecular design is one of nonlinear optimization, which can be carried out using a number of strategies. Specifically, we have shown how direct, inverse, and hybrid strategies can be employed. Conventional chemical strategies for optimization usually focus on limited regimes of molecular space in which the optimization problem is considerable simpler. It is hoped that these strategies can be used to determine classes of “optimum” structures, rather than minor variants within a class or subclass. Clearly, increased effort is needed to apply these methods to larger systems with appropriate constraints. Many technical challenges remain. For small systems the structure-property surfaces are relatively simple in form. However, as the number of atoms increases, inspection is not a viable strategy for determining optimized structures. Rather, strategies for sampling surfaces, perhaps via Monte Carlo techniques, seem essential for further progress.14 Finally, interpreting the chemical meaning of optimized Hamiltonians is challenging. Analysis of the derived bond orders and bond energies15 may provide a way forward. In spite of these challenges, we believe that application of the inverse strategy to molecular design could open up a number of areas of materials science to the rational design of new classes of chromophores. These methods could be of particular help
(1) (a) Marder, S. R.; Beratan, D. N.; Cheng, L.-T. Science 1991, 252, 103. (b) Kuhn, C. Synth. Met. 1991, 41-43, 3681. (c) Miller, J. S.; Epstein, A. J. Chem. Eng. News 1995, 73, 30. (d) Andre´, J.-M.; Delhalle, J.; Bredas, J.-L. Quantum chemistry aided design of organic polymers; World Scientific: New Jersey, 1992. (e) Aubry, S.; Quemerais, P. In LowDimensional Electronic Properties of Molybdenum Bronzes and Oxides; Schlenker, C., Ed.; Kluwer, Deventer, 1989. (f) Yoshizawa, K.; Hoffmann, R. Chem. Eur. J. 1995, 1, 403. (g) Kanis, D. R.; Ratner, M. A.; Marks, T. J. Chem. ReV. 1994, 94, 195. (2) (a) Li, Y. S.; van Daelen, M. A.; Wrinn, M.; King-Smith, D.; Newsam, J. M.; Delley, B.; Wimmer, E.; Klitsner, T.; Sears, M. P.; Carlson, G. A.; Nelson, J. S.; Allan, D. C.; Teter, M. P. J. Comput.-Aided Mater. Des. 1993, 1, 199. (b) Wimmer, E. J. Comput.-Aided Mater. Des. 1993, 1, 215. (3) Some areas in which inverse design strategies are being pursued include: (a) Caudill, L. F.; Rabitz, H.; Askar, A. InVerse Problems 1994, 10, 1099. (b) Newon, R. G. InVerse Schro¨ dinger Scattering in Three Dimensions; Springer Verlag: New York, 1989. (c) Dorren, H. J. S.; Muyzert, E. J.; Snieder, R. K. InVerse Problems 1994, 10, 865. (d) Yue, K.; Dill, K. A. Proc. Natl. Acad. Sci. U.S.A. 1992, 89, 4163. (e) Godzik, A.; Kolinski, A.; Skolnick, J. J. Mol. Biol. 1992, 227, 227. (f) Shapiro, M.; Brumer, P. Int. ReV. Phys. Chem. 1994, 13, 187. (g) Shen, L. Y.; Shi, S. H.; Rabitz, H. J. Phys. Chem. 1993, 97, 12114. (4) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes, The Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1990. (5) For simplicity we discuss this design problem in the framework of an effective Hamiltonian theory, such as the tight-binding Hu¨ckel theory. Such semiempirical schemes write the diagonal and off-diagonal Hamiltonian elements as a function of atomic orbital properties and overlap integrals. As such, one can make a rough mapping from the “ideal” Hamiltonian onto “real” atoms and molecules. These schemes are described in: (a) Pople, J. A.; Beveredge, D. L. Approximate Molecular Orbital Theory; McGraw Hill: New York, 1970. (b) Harriman, W. A. Electronic Structure and the Properties of Solids; Dover Press: New York, 1989. (6) Risser, S. M.; Beratan, D. N.; Marder, S. R. J. Am. Chem. Soc. 1993, 115, 7719. (7) (a) Frauenfelder, H. In Structure and Dynamics of Nucleic Acids, Proteins and Membranes; Clementi, E., Chin, S., Eds.; Plenum Press: New York, 1986. (b) Socci, N. D.; Onuchic, J. N. J. Chem. Phys. 1994, 101, 1519. (8) (a) Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M. P. Science 1983, 220, 671. (b) Christofides, N., Mingozzi, A., Toth, P., Sandi, C., Eds. Combinatorial Optimization; Wiley-Interscience: London, 1979. (9) Murnaghan, F. D. The Unitary and Rotation Groups; Spartan Books: Washington DC 1962. (10) Note that this factoring strategy is different from three-dimensional Euler angle strategies. Extension of the Euler strategy to higher dimensions is not directly performed. (11) In example 2 below, this strategy is straightforward because of the problem’s 1D nature. If the problem is 2D or 3D, the correspondence between wave functions and eigenvalues is not trivial; see: Courant, R.; Hilbert, D. Methods of Mathematical Physics; Interscience Publishers: New York, 1953; Vol. 2. This difficulty is avoided in the hybrid method of section II(c). (12) (a) Arfken, G. Mathematical Methods for Physicists, 3rd ed.; Academic Press: New York, 1985. (b) NAG subroutine, Numerical Algorithms Group (USA). (13) (a) Piela, L.; Kostrowicki, J.; Scheraga, H. A. J. Phys. Chem. 1989, 93, 3339. (b) Scheraga, H. A. In Abstracts of Papers, 210th National Meeting of the American Chemical Society, Chicago, IL, Fall 1995; American Chemical Society: Washington, DC, 1985; PHYS 097. (14) Risser, S., private communication. (15) (a) Salem, L. Molecular Orbital Theory of Conjugated Systems; Benjamin: London 1966. (b) Helbronner, E.; Bock, H. Das HMO-Modell und seine Anwendung; Verlag Chemie: Weinheim, 1968.
JP960518I
