《Theory and Practice of Direct Methods IN crystallography》

Contents1

1．Principles of Direct Methods of Phase Determination1

in Crystal Structure Analysis1

G.Allegra1

1.1．Introduction1

1.2．Spherical Symmetry of Atoms:Sayre's Equation4

1.3．Unitary and Normalized Structure Factors6

1.4．Karle-Hauptman Determinants7

1.5．Structure Invariants and Seminvariants9

1.6．Probability Theory12

1.7．Solving the Phase Problem for a Real Structure16

1.8．Refinement of Phases18

1.9．Possible Future Developments of Direct Methods19

References21

2．Definition of Origin and Enantiomorph and Calculation of|E| ValuesD.RogersPART Ⅰ:Definition of Origin and Enantiomorph23

2.1．Introduction23

2.2．Some Preliminaries24

2.3．Invariance32

2.4．Variation of Phase among the Laue-Related Refiections36

2.5．Defining the Origin43

2.5.1．Primitive Centrosymmetric Space Groups43

2.5.2．Nonprimitive Centrosymmetric Space Groups49

2.5.3．Noncentrosymmetric Space Groups50

2.5.4．Primitive Noncentrosymmetric Space Groups57

2.5.5．Nonprimitive Noncentrosymmetric Space GroupsZ69

2.6．Some Unusual Requirements of S/I Selectors71

2.7．In Conclusion76

Appendix 2A.177

Appendix 2A.280

PART Ⅱ:Calculation of|E|Values82

References91

3．Symbolic Addition and Multisolution Methods93

M.F.C.Ladd and R.A.Palmer93

3.1．Introduction93

3.2．Symbolic Addition:Centrosymmetric Case100

3.2.1．Phase Determination in Space Group Pī:Pyridoxal Phosphate Oxime Dihydrate100

3.3．Symbolic Addition:Noncentrosymmetric Case103

3.3.1．Phase Determination in Space Group P21:Tubercidin104

3.4．Advantages and Disadvantages of Symbolic Addition113

3.5．Multisolution Methods113

3.5.1．Introduction:Multisolution Philosophy and Brief Description of the Program MULTAN113

3.5.2．Centrosymmetric Case:Papaverine Hydrochloride118

3.5.3．Noncentrosymmetric Case:Methyl Warifteine and Dimethyl Warifteine118

3.5.4．Experience with Large Structures:Ribonuclease-Potassium Hexachloroplatinate127

3.6．Success Is Not Guaranteed140

3.6.1．Some Prerequisites for Success in Using Direct Methods140

3.6.2．Figures of Merit:a Practical Guide141

3.6.3．Signs of Trouble,and Past Remedies When the Structure Failed to Solve141

3.6.4．Comments on Molecular Scattering Factors142

3.7．More Recent Developments of the Multisolution Method:Magic Integers144

References149

4．Probabilistic Theory of the Structure Seminvariants151

Herbert Hauptman151

4.1．Major Goal151

4.2．Introduction152

4.3．Structure Invariants153

4.4．Structure Seminvariants155

4.5．The Structure Seminvariants Link the Observed Magnitudes|E| with the Desired Phases？158

4.6．Probabilistic Background159

4.7．Three-Phase Structure Invariant159

4.7.1．Space Group Pl159

4.7.2．Space Group Pī162

4.8．Four-Phase Structure Invariant(Quartet)162

4.8.1．Space Group Pl162

4.8.2．Space Group Pī168

4.9．The Neighborhood Principle172

4.10．More on Quartets:Higher Neighborhoods173

4.10.1．Third Neighborhoods of the Structure Invariant φ4＝φh＋φk＋φl＋φm174

4.10.2．Higher Neighborhoods175

4.10.3．Probability Distributions in Pl Derived from the Third(Thirteen-Magnitude)Neighborhoods176

4.10.4．Probability Distributions in Pī Derived from the Third (Thirteen-Magnitude)Neighborhoods181

4.11．Two-Phase Structure Seminvariants(Pairs)186

4.11.1．Space Group Pī187

4.11.2．Space Group P21188

4.11.3．Space Group P212121192

4.12．Concluding Remarks196

References196

5．Application of Calculated Cosine Invariants in Phase DeterminationWilliam Duax,Charles Weeks,Herbert Hauptman,George DeTitta,David Langs,Edward Green,and Gert Kruger196

5.1．Introduction199

5.2．Accuracy of Cosine Calculations201

5.3．Quartets,Quintets,and Triplets206

5.4．∑1 Cosines211

5.5．Pair Relationships213

5.6．Strong Enantiomorph Selection218

5.7．NQEST223

5.8．Automated Procedures226

5.9．Exercises226

References229

6．Phase Correlation with Calculated Cosine Invariants for Routine Structure AnalysisPaul T.Beurskens and Th.E.M.van den Hark229

6.1．Phase Correlation Procedure231

6.1.1．Definitions232

6.1.2．Selection of the Starting Set232

6.1.3．Phase Generation233

6.1.4．Correlation Equations233

6.1.5．End of the Phase Correlation Procedure234

6.1.6．Quadruples and Quartets in Relation to the Phase Correlation Procedure234

6.1.7．Dinaphtho［1,2-a;1'2'-h]anthracene235

6.2．Simple Cosine-Invariant Calculations241

6.2.1．Aminomalonic Acid243

6.2.2．Scaling of Calculated Triple Invariants244

6.2.3．Definition of KABhk Values244

6.3．Phase Correlation with Calculated Triple Invariants246

6.3.1．Centrosymmetric Space Groups247

6.3.2．Noncentrosymmetric Space Groups with Three Centric Projections248

6.3.3．Noncentrosymmetric Space Groups with One Centric Projection249

References253

7．Application of Direct Methods to Difference Structure FactorsPaul T.Beurskens and Th.E.M.van den Hark253

7.1．Introduction255

7.1.1．Some Applications of DIRDIF Procedures256

7.1.2．Two-Dimensional Wilson Plot257

7.2．Difference Structure Factors259

7.2.1．Probability Considerations261

7.2.2．Classification of Reflections262

7.3．Description of the Procedure264

7.3.1．Tangent Refinement264

7.3.2．Origin Specification267

7.3.3．Trichloro-bis(triethylphosphine)cobalt(Ⅲ)268

7.3.4．Heptahelicene269

7.4．Some Observations269

References270

8．Phase Extension and Refinement Using Convolutional and Related Equation SystemsDavid Sayre270

8.1．Introduction271

8.2．Outline of the Convolutional Equation Systems272

8.3．Outlines of the Phasing Methods Based on the Convolutional Equation Systems276

8.3.1．Tangent-Formula Refinement277

8.3.2．Density Modification Methods278

8.3.3．Least-Squares Phase Retinement280

8.3.4．Davies-Rollett Technique281

8.4．Comments on the Phasing Methods Based on the Convolutional Equation Systems281

8.5．Computational and Practical Aspects282

8.6．Summary284

References284

9．Maximum Determinant Method287

G.Tsoucaris287

9.1．Introduction287

9.2．Inequalities and Algebraic Properties of Determinants288

9.2.1．Unitary and Normalized Structure Factors288

9.2.2．Karle-Hauptman Determinants(1950)290

9.2.3．Determinants as a Function of Structure Invariants293

9.2.4．The Ellipsoid Representation of Inequalities,and Efficiency in Phase Determinationv296

9.3．Maximum Determinant Rule304

9.3.1．Definition304

9.3.2．Proof305

9.3.3．Eigenvectors and the Ellipsoid Representation312

9.3.4．Regression Equation318

9.4．Practical Applications323

9.4.1．Low-Order Determinants,and Determination of Structure Invariants323

9.4.2．Medium-Order Determinants,and Their Use in ab initio Structure Determination328

9.4.3．High-Order Determinants,and Their Use in Protein Structure Determination332

9.5．New Gram Determinants336

9.5.1．Use of Stereochemical Information-Known Fragments336

9.5.2．The"Moduli-Model-Phase"Determinant343

9.5.3．The Equiprobability Function in Direct Space343

Appendix 9A.1．Expression of the Hermitian Form Qm for m＝1,2,345

Appendix 9A.2．Joint Probability for All Structure Factors:Compound Probability Law347

Appendix 9A.3．Negative Triplet Determination348

Appendix 9A.4．Determination of the Sign Invariants of the Last Row of Am+1350

Appendix 9A.5．Hilbert Space and Gram Determinants357

Referencesv359

10．Molecular Replacement Method361

Patrick Argos and Michael G.Rossmann361

10.1．Introduction361

10.2．Preliminary Theoretical Considerations364

10.3．Rotation Function366

10.3.1．Fundamentals366

10.3.2．Reciprocal-Space Expression367

10.3.3．Matrix Algebra369

10.3.4．Symmetry372

10.3.5．Sampling and Background375

10.3.6．Locked Rotation Function and Klug Peaks375

10.3.7．Recognizing Known Fragments377

10.3.8．Fast Rotation Function379

10.4．Translation Problem381

10.4.1．Introduction381

10.4.2．Neither Structure is Known383

10.4.3．Positioning of a Known Molecular Structure386

10.4.4．Both Structures are Known388

10.4.5．Use of Heavy Atoms to Determine a Molecular Center388

10.4.6．Use of Packing Considerations389

10.5．Phase Determination389

10.5.1．Introduction389

10.5.2．Reciprocal-Space Equations390

10.5.3．Real-Space Molecular Replacement393

10.5.4．Equivalence of Real-and Reciprocal-Space Molecular Replacement395

10.6．Noncrystallographic Symmetry and Heavy-Atom Searches397

10.7．Applications of Molecular Replacement410

10.8．Conclusions412

References413

INDEX419