A General SCF Theory

We live in a molecular world, almost closed shell in nature, and for this reason Chemistry has been a science dealing with closed shell mol­ ecules. However, the high degree of experimental sophistication reached in the past decade has made more apparent the role of open shell structures in chemical research. A parallel phenomenon can be observed in the development of SCF theory, where closed shell molecular calculations at any level of complexity compose the main body of references which can be obtained in Quantum Chemistry today. Besides the linkage between experimental and theoretical behaviour, there are, obviously, other reasons which can be attached to a lack of molecular open shell calculations. Among others, there was no connec­ tionbetween closed or open shell theoretical treatments. In this manner, many computational features used by closed shell connoisseurs have not been extended to other computational areas. Since the work of Roothaan in 1960, the open shell molecular landscape has been, the­ oretically, a very closed one. Further development of SCF theory, which has led to an outburst of multiconfigurational procedures, has paid no, or very faint, attention to the interconnection between these SCF theory advanced features, the open shell framework and closed shell common practice. A good theoretical goal, generally speaking, and in particular inside SCF theory, may consist of a procedure which can be used to solve a given chemical problem, within the physical and approx­ imate limits of the theory.

I. Historical Review.- 1. The Open Shell Development.- 2. The Multiconfigurational Scheme.- II. Electronic Energy, Fock Operators and Coupling Operators.- 1. Introduction.- 2. General Energy Expression.- 3. General Coulomb and Exchange Operators.- 4. Energy and Lagrangian Variation: Fock Operators and Euler Equations.- 5. Coupling Operator.- 6. Null Gradient Coupling Operator Part.- 7. Lagrange multipliers Hermitean Condition Coupling Operator Part.- 8. LCAO Form of Coupling Operator.- 9. Simplified Energy Forms. Monoconfigurational Open Shell.- 10. Closed Shell.- 11. Corollary.- III. Eigenspace Manipulations.- 1. Introduction.- 2. General Formalism.- 3. Unconditional Convergence in SCF Procedures: Level Shift Techniques.- 3.1. The Ordering Principle.- 3.2. Level Shift Technique in a General SCF Procedure.- 3.3. Koopman’s Theorem and Level Shifted Closed Shell Fock Operators.- 3.4. Final Remarks.- IV. Multiconfigurational Structure of Monoconfigurational SCF Procedures.- 1. Monoconfigurational Energy and Euler Equations.- 2. Results.- 3. Final Remarks.- V. Paired Excitation Multiconfigurational SCF.- 1. Closed Shell MCPESCF Theory.- 2. Complete MCPE Energy Expression.- 3. Two Electron Systems.- 4. MCPESCF Theory with an Invariant Closed Shell.- 5. Open Shell PEMCSCF.- 6. Special Cases.- VI. SCF Perturbation Theory.- 1. Perturbation Theory.- 1.1. General Scheme.- 1.2. Orthonormalization Conditions.- 1.3. Eigenvalue Corrections.- 1.4. Wigner’s Theorem.- 1.5. Eigenvector Corrections.- 1.6. Alternative Formalism.- 2. Open Shell SCF Theory.- 2.1. Energy.- 2.2. Eigenvector Corrections.- 2.3. Procedure.- 2.4. Closed Shell SCF Perturbation Theory.- 3. Interaction of Two Molecules as an Application Example.- 3.1. General Background.- 3.2. Variational Equations.- 3.3. Perturbational Scheme.- 3.4. Nature of the Interaction Energy.- 3.5. Electrostatic Molecular Potential.- VII. General Theory for Two and Three Electron Systems.- 1. Two Electron Systems.- 1.1. Singlet States.- 1.2. Triplet States.- 2. Three Electron Systems.- 2.1. Doublet States. Case A.- 2.2. Doublet States. Case B.- 2.3. Quadruplet States.- VIII. Approximate SCF Theories.- 1. Atomic Orbital Representation.- 1.1. Introduction.- 1.2. Representation of AO’s.- 1.3. Representation of Charge Density: Mulliken’s Gross Atomic Populations as a Natural Way of Charge Partitioning.- 1.4. Expectation Values of Monoelectronic Operators.- 1.5. SCF Theory and Energy Partitioning into Atomic Contributions.- 1.6. General Remarks.- 2. Decomposition of the Electronic Repulsion Matrix.- 3. Empirical Approximate Methods.- 4. Model Potentials: Huzinaga’s Approach.- IX. Miscellaneous Remarks.- 1. A Synthetic Approach.- 2. The Concept of Shell.- 3. Symmetry.- 4. Optimization of Non-Linear Parameters.- 5. Generalized Brillouin’s Theorem and Off-Diagonal Hermitean Conditions on Lagrange Multipliers.- 6. Error Analysis.- 7. Mathematical Structure of SCF.- X. The Problem of the Helium Atom First Excited Singlet State.- 1. A Possible Solution.- 2. The “Triplet Catastrophe”.- 3. Further Analysis of the “The Triplet Catastrophe”.- 4. Some Results on Monoconfigurational He SCF.- XI. Applications.- 1. Introduction.- 2. SCF Study of Water: Ground and Excited States.- 3. Paired Excitation Calculation on Water.- 4. Formaldehyde.- 5. Magnesium Oxyde.- 6. Nitrogen Dioxyde.- 7. Methanol.- 8. Diimine.- 9. Methylenimine.- 10. Glycine.- 11. Excited States of Some Molecules with C=O and C=N Bonds: INDO Procedure.- Appendix A: Monoconfigurational State Parameters.- Appendix B: Slater Rules.- Appendix C: Multiconfigurational Fock Operators.- Suggested Reading.- Bibliographical Survey.- A. Open Shell SCF Theory.- B. Multiconfigurational SCF Theory.