Ab Initio and Semiempirical Quantum Chemistry

In order to eliminate any fundamental dependence on the assumed potential model, one must resort to the more complicated and time-consuming methods of quantum chemistry. Whereas clearly these techniques in theory provide the correct treatment of molecular and atomic systems, incomplete basis sets and limitations due to electron correlation or exchange for the methods capable of treating reasonably sized molecules can present considerable complications. In the present case, we are primarily interested in using quantum chemistry methods to evaluate structural geometries as determined by the classical deterministic (molecular mechanics and dynamics) and stochastic

FIGURE 14.5 Final structure of a 35 monomer (four tetrahedral defects) MEH-PPV molecule obtained through a sequences of continuum and explicit solvent molecular dynamics and mechanics simulations combined with simulated annealing.

methods (Monte Carlo) described earlier. The quantum methods also provide an avenue for computing electronic absorption and emission spectra for comparison to observed experimental luminescence spectroscopy. As the excited state structure depends on the ground state, it is extremely important to have the correct optimization of the structure for the ground state. A structure taken from other computations will most likely not be the optimal one for the representation of the wave function in quantum chemistry calculations and likewise computations of excited state properties will not conform to any meaningful result. As such we feel that it is necessary to go through some of the fundamental details of the various quantum chemistry methods so as to clearly define what we have done and why.

Quantum mechanical methods are most generally based on the wave function approach [45], although density functional theory (DFT), which determines the ground state electronic structure as a function of the electron density [46], offers an equivalently powerful method (see below). The wave function approach to electronic structure determination begins with the Schroedinger equation in which the Hamiltonian operator is composed of the kinetic energy of the nuclei and the electrons, the electrostatic (Coulombic) interactions between the nuclei and the electrons, and the internuclear and interelectronic repulsions. Simplifications are generally made by imposing the Born-Oppenheimer approximation, which assumes that the nuclei do not move, and hence have no kinetic energy. This removes the terms for the nuclear kinetic energy and the nuclear-nuclear repulsions become a constant that is simply added to the electronic energy to get the total energy of the system. In order to determine a practical representation of the unknown wave function, the many-electron wave function is replaced by a product of one-electron wave functions. The simplest such replacement, called the Hartree-Fock (HF) (or single determinant) wave function, involves a single determinant of products of one-electron functions, called spin orbitals. Each spin orbital is written as a product of a space part, F, describing the location of a single electron, and one of two possible spin parts, a or b. The space part is essentially a molecular orbital (MO), which can be occupied by only two electrons of opposite spin. By using a linear combination of atomic orbitals (LCAO) approximation (the molecular wave functions are composed of atomic wave functions, combined linearly) one obtains what are called the Roothaan-Hall equations. These equations can be solved numerically using a self-consistent iterative procedure, in which each MO is evaluated under the influence of an average potential field from the other electrons. The derived MO then contributes to the field used for the next MO, and the process repeats until it is internally consistent within specified limitations (SCF-HF).

The total HF energy is then given by

EHF _ ^nuclear ^ ^core _^ ^Coulomb ^ ^exchange (14 1)

The four terms are defined as follows: • Nuclear term nuclei ry ry

Core term basis functions

Coulomb term m n basis functions

Exchange term mn basis functions

m n where the density matrix, Pmn, the elements of which are the squares of the MO coefficients summed over all occupied orbitals is

Computations based on these approximations give quite good structures for molecules containing main group elements. However, even in the limit of a complete basis set, the HF energy is not equal to the experimental energy of a molecule, largely because of the error introduced by using the SCF model (electron correlation being one problem). These methods also scale somewhat poorly as a function of the number of basis functions ~N4, which tend to limit calculations to systems with less than 100 atoms. On the other hand, DFT, which is based on the Hohenberg-Kohn theorem, scales more like N3 and in theory accounts for electron correlation (although this assumes correct exchange-correlation func-tionals) [47]. In DFT, the minimum energy of a collection of electrons under the influence of an external field (the nuclei) is a function of the electron density. The energy includes the same nuclear, core, and Coulomb terms as the HF energy. However, the HF exchange energy is replaced by an exchange-correlation functional, EX (p), leading to the Kohn-Sham equation:

This unique functional, valid for all systems can be formally proved, but an explicit form of the potential has been difficult to define; hence the large number of DFT methods. Functional forms are often chosen based on which ones give the best fit to a certain body of experimental data, which makes DFT more like a semiempirical method.

One advantage of using electron density is that the integrals for Coulomb repulsion need to be done only over the electron density, which leads to the N3 scaling. The other advantage is that the use of electron density automatically includes at least some of the electron correlation. Among the simplest models are those called the local density models, such as the SVWN (Slater, Vosko, Wilk, Nusair) [48] functional. The basis of these models is the assumption of a many-electron gas of uniform density. This model is generally not expected to be satisfactory for molecular systems, in which the electron density is nonuniform. However, for band structure it is quite satisfactory. Substantial improvement can often be obtained by introducing explicit dependence on the gradient of the electron density, as well as the density itself. Such procedures are called gradient-corrected, or nonlocal DFT models. The most popular of these models are the so-called BP (Becke and Perdew) [49], BLYP (Becke, Lee, Yang, and Parr) [50], and PBE (Perdew, Burke, and Ernzerhof) [51]. Another class of DFT models combines the exact HF exchange with a DFT exchange term, and adds a correlation functional [52]. In general, DFT-based methods can treat larger systems and even provide a route to achieve O(N) scaling. However, the current state-of-the-art is still somewhat premature to be used as a black box.

Semiempirical MO methods lie somewhere between classical molecular mechanics and ab initio quantum mechanics methods [53]. This approach makes use of a number of experimentally determined parameters but like fundamental ab initio quantum methods is based on a SCF-HF solution with atomic orbital (AO) basis functions. Semiempirical models use s-orbitals and the px and py orbitals for the valence shell. The remaining part of each atom is treated as a core, with a net or effective charge equal to the atomic number Z minus the number of inner shell electrons. The mathematical representation of each of the four basic orbitals, used in constructing the LCAO, is a representation that reflects the spatial distribution of the electrons occupying the orbitals. The integrals are divided into two sets, H and S, according to whether they contained the Hamiltonian operator. Semiempirical methods represent the H-type integrals as a sum of five terms: (1) one-center, one-electron integrals, which represent the sum of the kinetic energy of an electron in an AO on atom X and its potential energy due to attraction to its own core; (2) one-center, two-electron repulsion integrals; (3) two-center, one-electron core resonance integrals; (4) two-center, one-electron attractions between an electron on atom X and the core of atom Y and (5) two-center, two-electron repulsion integrals.

The integrals of types (1) and (2) are replaced by numerical values obtained by fitting spectroscopic values of electron energies in various valence states. Because these parameters are based on experimental data from real molecules, in which electron correlation is a fact of nature, some correlation is built into semiempirical methods. This makes up, in part, for the failure of semiempirical methods to consider correlation explicitly. Integrals of type (3) represent the main contribution to the bonding energy of the molecule. Semiempirical methods treat these as proportional to the overlap integral, S:

where fx is an adjustable parameter. Note that none of these terms are set to zero as in Huckel theory, however, they do not contain any explicit dependence on interatomic distance. A modified electrostatic treatment is used to evaluate the integrals of type (4). They are taken as proportional to eZf/r, with an adjustable parameter for the proportionality constant; they also include a term of the form exp(— aR), where a is an adjustable parameter. This function is included to ensure that the net repulsion between neutral atoms vanishes as their separation goes to infinity.

The remaining integrals, type (5), represent the energy of interaction between the charge distribution at atom X and that at atom Y They are calculated as the sum of the over all multipole interactions, again with adjustable parameters. STOs are used to represent the spatial distribution of the charges. The overlap integrals, S, are evaluated analytically, however, the orbital exponents are treated as adjustable parameters.

Core repulsions (the core of one atom interacting with the core of the next) are the remaining item to be evaluated. This is done with two center repulsion integrals, and the cores are represented as

Gaussians. In order to reduce excessive repulsions at large atomic separations, which arise in a more classical electrostatic treatment, attractive Gaussians are added to the repulsive ones for most elements. Finally, the total energy of the molecule is represented as the sum of the electronic energy (net negative) and the core repulsions (net positive).

With this general structure of the model completed, a so-called training set of molecules is selected, chosen to cover as many types of bonding situations as possible. A nonlinear least square optimization procedure is applied with the values of the various adjustable parameters as variables and a set of measured properties of the training set as constants to be reproduced. The measured properties include heats of formation, geometrical variables, dipole moments, and first ionization potentials.

Depending on the choice of the training sets, the exact number of types of adjustable parameters, and the mode of fitting to experimental properties, different semiempirical methods have been developed, ranging from the AM1 (Austin Model 1) method of M.J.S. Dewar and the PM3 (Parameter Model 3) method of J.J.P. Stewart, to variants of the older MNDO models.

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