Theory and Practice of Direct Methods in Crystallography

Direct methods of crystal structure determination are usually associated with techniques in which phases for a set of structure factors are determined from the corresponding experimental amplitudes by probabilistic calcula­ tions. It is thus implied that such ab initio phase calculations do not require a knowledge of atomic positions, and this basis distinguishes direct methods from other techniques for structure determination. An acceptably wider interpretation of the term direct methods leads to other important applica­ tions involving, inter alia, the use of heavy atoms, resolution-limited phase data for large molecules, rotation functions, and Fourier series. These topics are discussed in the later chapters of this book. Although some earlier theoretical investigations were made by Harker and Kaspar, direct methods may be considered to have begun around the year 1950. Important landmarks in the development of the subject include the book by Hauptmann and Karle, The Centrosymmetric Crystal (1953), the definitive paper by Karle and Karle in Acta Crystallographica (1966), and the recent (1978) sophisticated program package MULTAN 78 produced mainly by Germain, Main, and Woolfson. Woolfson's book, Direct Methods in Crystallography, was published in 1961, but because of the rapid progress in direct methods, much of it soon became outmoded. It is interesting to note that direct methods nearly came into being many years earlier. Certainly the E2 relationship was used implicitly by Lonsdale in 1928 in determining the crystal structure of hexamethylbenzene.

1. Principles of Direct Methods of Phase Determination in Crystal Structure Analysis.- 1.1. Introduction.- 1.2. Spherical Symmetry of Atoms: Sayre’s Equation.- 1.3. Unitary and Normalized Structure Factors.- 1.4. Karle-Hauptman Determinants.- 1.5. Structure Invariants and Seminvariants.- 1.6. Probability Theory.- 1.7. Solving the Phase Problem for a Real Structure.- 1.8. Refinement of Phases.- 1.9. Possible Future Developments of Direct Methods.- References.- 2. Definition of Origin and Enantiomorph and Calculation of |E| Values.- I: Definition of Origin and Enantiomorph.- 2.1. Introduction.- 2.2. Some Preliminaries.- 2.3. Invariance.- 2.4. Variation of Phase among the Laue-Related Reflections.- 2.5. Defining the Origin.- 2.5.1. Primitive Centrosymmetric Space Groups.- 2.5.2. Nonprimitive Centrosymmetric Space Groups.- 2.5.3. Noncentrosymmetric Space Groups.- 2.5.4. Primitive Noncentrosymmetric Space Groups.- 2.5.5. Nonprimitive Noncentrosymmetric Space Groups.- 2.6. Some Unusual Requirements of S/I Selectors.- 2.7. In Conclusion.- Appendix 2A.1.- Appendix 2A.2.- II: Calculation of |E| Values.- References.- 3. Symbolic Addition and Multisolution Methods.- 3.1. Introduction.- 3.2. Symbolic Addition: Centrosymmetric Case.- 3.2.1. Phase Determination in Space Group $$P\mathop 1\limits^ - $$: Pyridoxal Phosphate Oxime Dihydrate.- 3.3. Symbolic Addition: Noncentrosymmetric Case.- 3.3.1. Phase Determination in Space Group P21: Tubercidin.- 3.4. Advantages and Disadvantages of Symbolic Addition.- 3.5. Multisolution Methods.- 3.5.1. Introduction: Multisolution Philosophy and Brief Description of the Program MULTAN.- 3.5.2. Centrosymmetric Case: Papaverine Hydrochloride.- 3.5.3. Noncentrosymmetric Case: Methyl Warifteine and Dimethyl Warifteine.- 3.5.4. Experience with Large Structures: Ribonuclease-Potassium Hexachloroplatinate.- 3.6. Success Is Not Guaranteed.- 3.6.1. Some Prerequisites for Success in Using Direct Methods.- 3.6.2. Figures of Merit: a Practical Guide.- 3.6.3. Signs of Trouble, and Past Remedies When the Structure Failed to Solve.- 3.6.4. Comments on Molecular Scattering Factors.- 3.7. More Recent Developments of the Multisolution Method: Magic Integers.- References.- 4. Probabilistic Theory of the Structure Seminvariants.- 4.1. Major Goal.- 4.2. Introduction.- 4.3. Structure Invariants.- 4.4. Structure Seminvariants.- 4.5. The Structure Seminvariants Link the Observed Magnitudes |E| with the Desired Phases ?.- 4.6. Probabilistic Background.- 4.7. Three-Phase Structure Invariant.- 4.7.1. Space Group P1.- 4.7.2. Space Group $$P\mathop 1\limits^ - $$.- 4.8. Four-Phase Structure Invariant (Quartet).- 4.8.1. Space Group P1.- 4.8.2.Space Group $$P\mathop 1\limits^ - $$.- 4.9. The Neighborhood Principle.- 4.10. More on Quartets: Higher Neighborhoods.- 4.10.1. Third Neighborhoods of the Structure Invariant $${\phi _4} = {\phi _h} + {\phi _k} + {\phi _l} + {\phi _m}$$.- 4.10.2.Higher Neighborhoods.- 4.10.3. Probability Distributions in PI Derived from the Third (Thirteen-Magnitude) Neighborhoods.- 4.10.4. Probability Distributions in $$P\mathop 1\limits^ - $$ Derived from the Third (Thirteen-Magnitude) Neighborhoods.- 4.11. Two-Phase Structure Seminvariants (Pairs).- 4.11.1. Space Group $$P\mathop 1\limits^ - $$.- 4.11.2. Space Group P21.- 4.11.3. Space Group P212121.- 4.12. Concluding Remarks.- References.- 5. Application of Calculated Cosine Invariants in Phase Determination.- 5.1. Introduction.- 5.2. Accuracy of Cosine Calculations.- 5.3. Quartets, Quintets, and Triplets.- 5.4. ?1 Cosines.- 5.5. Pair Relationships.- 5.6. Strong Enantiomorph Selection.- 5.7. NQEST.- 5.8. Automated Procedures.- 5.9. Exercises.- References.- 6. Phase Correlation with Calculated Cosine Invariants for Routine Structure Analysis.- 6.1. Phase Correlation Procedure.- 6.1.1. Definitions.- 6.1.2. Selection of the Starting Set.- 6.1.3. Phase Generation.- 6.1.4. Correlation Equations.- 6.1.5. End of the Phase Correlation Procedure.- 6.1.6. Quadruples and Quartets in Relation to the Phase Correlation Procedure.- 6.1.7. Dinaphtho [1,2-a; 1’,2’-h]anthracene.- 6.2. Simple Cosine-Invariant Calculations.- 6.2.1. Aminomalonic Acid.- 6.2.2. Scaling of Calculated Triple Invariants.- 6.2.3. Definition of KABhk Values.- 6.3. Phase Correlation with Calculated Triple Invariants.- 6.3.1. Centrosymmetric Space Groups.- 6.3.2. Noncentrosymmetric Space Groups with Three Centric Projections.- 6.3.3. Noncentrosymmetric Space Groups with One Centric Projection.- References.- 7. Application of Direct Methods to Difference Structure Factors.- 7.1. Introduction.- 7.1.1. Some Applications of DIRDIF Procedures.- 7.1.2. Two-Dimensional Wilson Plot.- 7.2. Difference Structure Factors.- 7.2.1. Probability Considerations.- 7.2.2. Classification of Reflections.- 7.3. Description of the Procedure.- 7.3.1. Tangent Refinement.- 7.3.2. Origin Specification.- 7.3.3. Trichloro-bis(triethylphosphine)cobalt(III).- 7.3.4. Heptahelicene.- 7.4. Some Observations.- References.- 8. Phase Extension and Refinement Using Convolutional and Related Equation Systems.- 8.1. Introduction.- 8.2. Outline of the Convolutional Equation Systems.- 8.3. Outlines of the Phasing Methods Based on the Convolutional Equation Systems.- 8.3.1. Tangent-Formula Refinement.- 8.3.2. Density Modification Methods.- 8.3.3. Least-Squares Phase Refinement.- 8.3.4. Davies-Rollett Technique.- 8.4. Comments on the Phasing Methods Based on the Convolutional Equation Systems.- 8.5. Computational and Practical Aspects.- 8.6. Summary.- References.- 9. Maximum Determinant Method.- 9.1. Introduction.- 9.2. Inequalities and Algebraic Properties of Determinants.- 9.2.1. Unitary and Normalized Structure Factors.- 9.2.2. Karle-Hauptman Determinants (1950).- 9.2.3. Determinants as a Function of Structure Invariants.- 9.2.4. The Ellipsoid Representation of Inequalities, and Efficiency in Phase Determination.- 9.3. Maximum Determinant Rule.- 9.3.1. Definition.- 9.3.2. Proof.- 9.3.3. Eigenvectors and the Ellipsoid Representation.- 9.3.4. Regression Equation.- 9.4. Practical Applications.- 9.4.1. Low-Order Determinants, and Determination of Structure Invariants.- 9.4.2. Medium-Order Determinants, and Their Use in ab initio Structure Determination.- 9.4.3. High-Order Determinants, and Their Use in Protein Structure Determination.- 9.5. New Gram Determinants.- 9.5.1. Use of Stereochemical Information—Known Fragments.- 9.5.2. The “Moduli-Model-Phase” Determinant.- 9.5.3. The Equiprobability Function in Direct Space.- Appendix 9A.1. Expression of the Hermitian Form Qm for m = 1,2.- Appendix 9A.2. Joint Probability for All Structure Factors: Compound Probability Law.- Appendix 9A.3. Negative Triplet Determination.- Appendix 9A.4. Determination of the Sign Invariants of the Last Row of.- Appendix 9A.5. Hilbert Space and Gram Determinants.- References.- 10. Molecular Replacement Method.- 10.1. Introduction.- 10.2. Preliminary Theoretical Considerations.- 10.3. Rotation Function.- 10.3.1. Fundamentals.- 10.3.2. Reciprocal-Space Expression.- 10.3.3. Matrix Algebra.- 10.3.4. Symmetry.- 10.3.5. Sampling and Background.- 10.3.6. Locked Rotation Function and Klug Peaks.- 10.3.7. Recognizing Known Fragments.- 10.3.8. Fast Rotation Function.- 10.4. Translation Problem.- 10.4.1. Introduction.- 10.4.2. Neither Structure is Known.- 10.4.3. Positioning of a Known Molecular Structure.- 10.4.4. Both Structures are Known.- 10.4.5. Use of Heavy Atoms to Determine a Molecular Center.- 10.4.6. Use of Packing Considerations.- 10.5. Phase Determination.- 10.5.1. Introduction.- 10.5.2. Reciprocal-Space Equations.- 10.5.3. Real-Space Molecular Replacement.- 10.5.4. Equivalence of Real- and Reciprocal-Space Molecular Replacement.- 10.6. Noncrystallographic Symmetry and Heavy-Atom Searches.- 10.7. Applications of Molecular Replacement.- 10.8. Conclusions.- References.