Semiempirical Methods of Electronic Structure Calculation

If one reflects upon the range of chemical problems accessible to the current quantum theoretical methods for calculations on the electronic structure of molecules, one is immediately struck by the rather narrow limits imposed by economic and numerical feasibility. Most of the systems with which experimental photochemists actually work are beyond the grasp of ab initio methods due to the presence of a few reasonably large aromatic ring systems. Potential energy surfaces for all but the smallest molecules are extremely expensive to produce, even over a restricted group of the possible degrees of freedom, and molecules containing the higher elements of the periodic table remain virtually untouched due to the large numbers of electrons involved. Almost the entire class of molecules of real biological interest is simply out of the question. In general, the theoretician is reduced to model systems of variable appositeness in most of these fields. The fundamental problem, from a basic computational point of view, is that large molecules require large numbers of basis functions, whether Slater­ type orbitals or Gaussian functions suitably contracted, to provide even a modestly accurate description of the molecular electronic environment. This leads to the necessity of dealing with very large matrices and numbers of integrals within the Hartree-Fock approximation and quickly becomes both numerically difficult and uneconomic.

1. Hückel Theory and Topology.- 1. Introduction.- 2. Equivalence between Hiickel Theory and the Graph Spectral Theory of Conjugated Molecules.- 3. Two-Color Problem in Hiickel Theory.- 4. Relationship between the Topology of Conjugated Systems and Their Corresponding Characteristic Polynomials.- 4.1. Characteristic Polynomial of a Conj ugated Molecule.- 4.2. Coulson-Sachs Graphical Method for the Enumeration of Coefficients of the Characteristic Polynomial.- 4.3. Summary of Some Results Obtained from the Coulson-Sachs Method.- 5. Topological Formulas for Hiickel Energy and ?-Resonance Energy.- 5.1. Topological Formula for ?-ElectronEneigy.- 5.2. Topological Formula for ?-Resonance Energy.- 6. Conclusions.- References.- 2. The Neglect-of-Differential-Overlap Methods of Molecular Orbital Theory.- 1. Background.- 1.1. Methodology.- 1.2. The Central Field Approximation and the Self-Consistent Field Procedure.- 1.3. The Form of the Basis Set.- 1.4. The ZDO Approximation.- 2. The NDO Methods.- 2.1. The CNDO Methods.- 2.2. The (M)INDO Methods.- 2.3. The (P)NDDO Methods.- References.- 3. The PCILO Method.- 1. Main Features of the PCILO Method.- 1.1. Advantages of the Localized MOs.- 1.2. Perturbative CI.- 1.3. Keeping the Simplicity of the CNDO Hamiltonian at the CI Level.- 1.4. Comparison with the Valence Bond Method.- 2. Derivation of the PCILO-CNDO Energy Contributions.- 2.1. Choice of the Localized MOs.- 2.2. The Fully Localized Determinant Energy.- 2.3. Second-Order Contributions.- 2.4. Third-Order Contributions.- 2.5. Improvement of Bond Polarities.- 3. Efficiency and Limits of the Method; Applications and Extensions.- 3.1. Nature of Possible Applications.- 3.2. Time and Memory Requirements; the Differential Scheme.- 3.3. Limitations of the Method.- 3.4. Brief Review of Applications to Ground-State Conformational Problems.- 3.5. Extensions of the Method.- 4. Concluding Remarks.- References.- 4. The X? Method.- 1. Introduction.- 1.1. The Origins of the Xa Method.- 1.2. Early Applications of the Method.- 1.3. Advantages and Disadvantages.- 2. Derivation of the Equations.- 2.1. The Energy Functional.- 2.2. The Slater Transition State.- 2.3. The Virial and Hellmann-Feynman Theorems.- 2.4. The Choice of Parameters.- 2.5. The Spin-Polarized and Relativistic Modifications.- 3. Applications of the Method.- 3.1. Atomic Calculations.- 3.2. Molecular Calculations.- 4. Comparison with Other Methods.- References.- 5. The Consistent Force Field and Its Quantum Mechanical Extension.- 1. Introduction.- 1.1. Efficiency.- 1.2. Reliability.- 1.3. Flexibility.- 2. Empirical Potential Functions.- 3. The Consistent Force Field (CFF) Method.- 3.1. The Philosophy of the CFF Method.- 3.2. The Refinement of the Potential Function Parameters.- 3.3. The Advantage of the Cartesian Representation.- 4. Quantum Mechanical Extension of the CFF Method to Ground and Excited States of Conjugated Molecules.- 4.1. Potential Surf aces for Conjugated Molecules.- 4.2. The Refinement of the Empirical Integrals.- 5. Applications.- 5.1. Energies, Conformations, and Vibrations of Large Molecules.- 5.2. Crystal Packing, Crystal Geometry, Lattice Dynamics, and Excimer Formation.- 5.3. Excited-State Geometries, Vibronic Interactions, and Photochemistry.- 5.4. Resonance Raman Intensities of Biologically Important Molecules.- 5.5. Classical Trajectories and Molecular Dynamics.- 6. Concluding Remarks.- References.- 6. Diatomics-in-Molecules.- 1. Introduction.- 2. Formulationof the Method.- 2.1. Molecular Energies and Wave Functions.- 2.2. Polyatomic Basis Functions.- 2.3. Partitioning of the Hamiltonian.- 2.4. Fundamental Approximation of DIM.- 2.5. Ab Initio DIM Theory.- 2.6. Semiempirical DIM Theory.- 3. Application of the Method.- 3.1. Selection of Basis Functions.- 3.2. Spin Coupling.- 3.3. Fragment Information.- 3.4. Overlap.- 4. Assessmentof the Method.- 4.1. Practicality.- 4.2. Analysis of Basic Approximations.- 4.3. Comparison with Accurate Results.- 4.4. Transferability.- 5. Properties Other Than Energy.- 6. Polyatomics-in-Molecules.- 7. Conclusions.- References.- 7. Theoretical Basis for Semiempirical Theories.- 1. Introduction.- 1.1. Division between Semiempirical and Ab Initio Fields.- 1.2. Need for a Theoretical Basis of Semiempirical Theories.- 2. Semiempirical Theories: Background.- 2.1. Traditional Formulation.- 2.2. Ambiguities and Difficulties.- 2.3. Earlier Derivations.- 3. The True Effective Valence Shell Hamiltonian.- 3.1. Basic Concepts.- 3.2. Derivation of ?v.- 3.3. Properties of ?v.- 4. Extraction of True Parameters.- 4.1. The True Parameters.- 4.2. Nonclassical Terms.- 4.3. Dynamic Variable Electronegativity.- 4.4. Properties Other Than Energies.- 4.5. The Chemical Orbitals.- 4.6. Extraction of the ??.- 5. Approximate Evaluation of True Parameters.- 5.1. Ab Initio Evaluation of the Correlation Parts of ?v.- 5.2. Difïuseness of ?* Valence States.- 5.3. ?v for Twisted Olefins.- 6. Model Pseudopotentials.- 6.1. The Usual Pseudopotential Equations.- 6.2. Exact Equations for the Valence Electrons.- 6.3. The Many-Electron Case.- 7. Discussion.- References.- Author Index.- Molecule Index.