Advances in Density Functional Theory (eBook)

Jorge M. Seminario (Herausgeber)

eBook Download: PDF

1998 | 1. Auflage
398 Seiten
Elsevier Science (Verlag)
978-0-08-058258-0 (ISBN)

Infos

Quantum mechanics can describe the detailed structure and behavior of matter, from electrons, atoms, and molecules, to the whole universe. It is one of the fields of knowledge that yield extraordinary precessions, limited only by the computational resources available. Among these methods is density functional theory (DFT), which permits one to solve the equations of quantum mechanics more efficiently than with any related method.
The present volume represents the most comprehensive summary currently available in density functional theory and its applications in chemistry from atomic physics to molecular dynamics. DFT is currently being used by more than fifty percent of computational chemists.

Inhalt

Front Cover 1
Advances in Quantum Chemistry: Density Functional Theory 4
Copyright Page 5
Contents 6
Contributors 12
Preface 16
Chapter 1. Why Density-Gradient Corrections Improve Atomization Energies and Barrier Heights 20
I. Energy Functionals and Chemical Energetics 20
2. Qualitative Explanations for Functional Performance 22
3. Density Parameters and Their Averages 22
4. Quantitative Explanation of Functional Performance 25
References 27
Chapter 2. Second-Order Relations Involving Correlation Energy and Its Functional Derivative 30
1. Introduction 31
2. Definitions 33
3. Different Correlation Energy Functionals 36
4. Identification of lim. EQC[nHF.] 38
5. Numerical Results 39
6. Relationships Involving the High-Density Limit of the Correlation Potential 41
7. Closing Remarks 45
References 46
Chapter 3. Asymptotic Properties of the Optimized Effective Potential 50
1. Introduction 50
2. The OEP Method 51
3. Approximation of Krieger, Li, and Iafrate 63
References 66
Chapter 4. Recent Developments in the Local-Scaling Transformation Version of Density Functional Theory 68
1. Introduction 68
2. Kinetic and Exchange Energy for Atoms in LS-DFT 69
3. Electronic Correlation in LS-DFT 79
4. Discussion 85
References 87
Chapter 5. In Search of the Correlation Potential 90
1. Introduction 91
2. Reference-State Density Functional Theory 92
3. Local Response Model 96
4. The Bound-Free Correlation Potential 98
5. Conclusions 99
References 101
Chapter 6. The n-Particle Picture and the Calculation of the Electronic Structure of Atoms, Molecules, and Solids 104
1. Introduction 104
2. Basic Notions 107
3. Generalization of the Hartree-Fock Approximation 113
4. Generalization of DFT 117
5. Discussion 121
References 122
Chapter 7. Gradient-Corrected Exchange Potential Functional with the Correct Asymptotic Behaviour 124
1. Introduction 125
2. Towards a Hierarchy in the Conditions Fulfilled by the Functionals 128
3. Importance of the Asymptotic Behaviour for the Potential 132
4. Calculation Details 134
5. Results and Discussion 135
6. Concluding Remarks 141
References 144
Chapter 8. Auxiliary Field Representation of Fermion Kinetic Energy Density Functional 150
1. Introduction 150
2. Background 151
3. Free Fermions, General 154
4. Extended Generating Functionals 157
5. F{nv} in One Dimension 159
6. Semi-classical Viewpoint 161
7. Explicit Functionals 164
8. Prospects 167
References 169
Chapter 9. Using the Exact Kohn-Sham Exchange Energy Density Functional and Potential to Study Errors Introduced by Approximate Correlation Functionals 170
1. Introduction 171
2. KLI Method and Atomic Energies 172
3. Gradient Corrections in Crystals 176
4. Conclusions 181
References 183
Chapter 10. Quantum Chemical Molecular Dynamics 186
1. Introduction 186
2. Forces and Fractional Occupation Numbers 188
3. Collinear Ozone Collisions 191
4. Collinear (NO)12 Collisions 197
5. Halide Photodissociation 200
6. Conclusions 203
References 204
Chapter 11. A Quantum Monte Carlo Approach to the Adiabatic Connection Method 208
1. Introduction 209
2. The Adiabatic Connection 210
3. Quantum Monte Carlo Realization 213
4. Cosine-Wave Jellium 215
5. Results and Discussion 218
References 225
Chapter 12. Full Potential Linearized-Augmented-Plane-Wave Calculations for 5d Transition Metals Using the Relativistic Generalized Gradient Approximation 228
1. Introduction and Summary of Results 228
2. Theoretical Background 230
3. Computational Details 233
4. Results 234
References 239
Chapter 13. Interatomic Force Constants in Periodic Solids from Density Functional Perturbation Theory 244
1. Introduction 244
2. Formalism 245
3. Interatomic Force Constants in Ionic and Covalent Solids 250
References 256
Chapter 14. Recent Developments in the Electronic Structure of Metal Surfaces 260
1. Introduction 261
2. Physical Interpretation of Kohn-Sham Theory 262
3. Asymptotic Structure in Vacuum Region 269
4. Structure of the Pauli and Correlation-Kinetic Components of Kohn-Sham Exchange Potential 276
5. Construction of Approximate Kohn-Sham Exchange Energy Functional and Derivative with Exact Asymptotic Structure 283
6. Concluding Remarks 287
References 288
Chapter 15. Density Functional Orbital Reactivity Indices: Fundamentals and Applications 292
1. Introduction 293
2. Method 293
3. Computational Details 301
4. Results and Discussion 302
5. Conclusions 309
References 309
Chapter 16. Density Functional Calculation of Reaction Energetics: Application to Alkyl Azide Decomposition 312
1. Introduction 312
2. Density Functional Calculations of Energies and Enthalpies of Reaction 314
3. An Application: Organic Azide Decomposition 314
4. Summary 319
References 320
Chapter 17. Density Functional Theory: A Source of Chemical Concepts and a Cost-Effective Methodology for Their Calculation 322
1. Introduction 324
2. Calculational vs. Conceptual DFT 324
3. Calculation of DFT-Based Quantities: Electronegativity and Hardness 329
4. Rationalizing Chemical Interactions within the HSAB Principle 336
5. Conclusions 343
References 344
Chapter 18. Pure and Mixed Pb Clusters of Interest for Liquid Ionic Alloys 348
1. Introduction 348
2. Pbn Clusters: Computational Details 349
3. Pbn Clusters: Results 350
4. Mixed Alkali–Pb Clusters 358
5. Study of Na6Pb Using the GGA 363
6. Summary 365
References 365
Chapter 19. Density Functional Theory in Catalysis: Activation and Reactivity of a Hydrocarbon Molecule on a Metallic Active Site 368
1. Introduction 368
2. Methodological Remarks 369
3. Catalytic Issues 370
4. DFT Description of the Interaction of a Methane Molecule with Active Sites 373
5. Discussion and Conclusions 382
References 385
Chapter 20. Recent Developments in High-Precision Computational Methods for Simple Atomic and Molecular Systems 388
1. Introduction 388
2. Variational Methods 389
3. Wave Functions 391
4. Helium 392
5. Lithium 396
6. Beryllium and Onward 399
7. Simple Molecules 402
8. Summary 403
References 404
Index 408

Textauszug

Preface

Jorge M. Seminario, Columbia, South Carolina

As the millennium comes to a close, a striking area of knowledge has emerged as the most promising for explaining the behavior of all known matter. This area is quantum mechanics. As experience teaches us, it is not wise to say that this is the final theory; nevertheless, we are witnessing how the equations of quantum mechanics, which predict the behavior of nature, can be used to design matter according to our needs and goals. Quantum mechanical equations can describe the detailed structure and behavior of matter, from electrons, atoms, and molecules to the whole universe. Quantum mechanics is an area of knowledge that yields the most remarkable precision. This precision is limited only by the computational resources available. For instance, the energy of helium-like atoms can be calculated with a precision of about 15 decimal places. Comparatively speaking, this is equivalent to determining the distance between one point in New York and another in Paris with a tolerance smaller than 10 Å. Such precision is possible when the nonrelativistic Schrödinger equation is sufficiently precise, a condition that covers a large part of chemistry.

Quantum mechanics has become a great tool for chemistry. For this reason, the methods of quantum mechanics used in chemistry have been grouped into a field called quantum chemistry, as in the title of this serial. The techniques of quantum chemistry were developed at a tremendous rate by the combined efforts of pure quantum theorists, application specialists, and scientific programmers, along with feedback from numerous precision experiments. These techniques have become routine in most universities worldwide, and courses on quantum chemistry at all levels are part of their curricula.

Solving the Schrödinger equation for chemical and chemistry-related applications from first principles has been the goal of two communities. Initially working separately, the standard ab initio and the density functional theory (DFT) groups had to deal with similar problems. In most cases, applications in chemistry require specific constraints on the methodologies to be used. For instance, it is necessary that methods be size-consistent or size-extensive, i.e., that systems of different size be treated with the same footing. In practical terms, this means that errors, if not zero, should in some way be proportional to the size of the system so that energy differences can take advantage of error cancellations. If a system only needs to be analyzed individually, this requirement may be irrelevant. A parallel and perhaps much more important requirement for solving the Schrödinger equation is that the chosen method not use ad hoc parameters. This is a very important requirement, of special significance in the design of new materials where experimental information is scarce or difficult to obtain; unfortunately, it is very challenging to fulfill. Sometimes a very small empirical correction can avoid a large computational process. Typical examples of this correction in ab initio methods are the high-level correction (HLC) in the G1 and G2 methods and the ~0.9 correction to the zero-point vibrational energies. Other common corrections are based on criteria of energy partitions; these include the basis set superposition error correction and the basis set incompleteness corrections. Alternative criteria are given by the need for variational methods, implying that the final energies are always upper bounds of the correct energy. This implies that an improvement in the basis set will always yield an energy closer to the exact one. Finally, in a practical sense, a very good compromise between precision and cost of the calculation is highly desired. Usually the cost of a calculation is determined by the scaling of the CPU time with respect to the size of the calculation and expressed in the form Nm, where N is an indicator of the size of the calculation (number of basis sets, number of electron, atoms, etc.) An N7 method will require 128 times more CPU resources if the size of the system is doubled. In practice, such a formal scaling is reduced by 1 or 2 units. For instance, the formal scaling for the HF method is N4, whereas practical calculations tend to scale close to N2.

The standard ab initio methods solve the Schrödinger equation in a direct way. They assume a wavefunction, which is expanded on a basis set and able to approach the ultimate solution in several ways. Unfortunately, this has a tremendous cost. These methods can initially use a single reference or a multi-reference wavefunction as a starting point. In the Hartree–Fock approximation, historically called the self-consistent method (SCF), the wavefunction is a Slater determinant composed of molecular orbitals representing the electrons in the molecule. These electrons interact with the mean field of the other electrons and the nuclei. Improvements to this approximation, which recognize that electrons actually do not interact with the mean field of other electrons but with each one individually, are performed with methods such as truncated configuration interaction, quadratic configuration interaction, Møller–Plesset perturbation theory, and coupled cluster theory. All these methods tend to obtain better and better solutions to the Schrödinger equation but simultaneously demand larger and larger basis sets in order to take advantage of the higher degree of treatment of the correlation energy, which is defined as the error of calculation with a single determinant, i.e., with the HF approximation. If a large enough basis set is used, a full configuration interaction method will yield the exact solution to the Schrödinger equation.

Alternatively, the convergence toward the precise solution of the Schrödinger equation can be accelerated by using a multireference approach. There is a corresponding multireference (MR) approach to each of the single reference approaches; these are named MR-HF or MR-SCF, MR-CI, MR-MP, and MRCC. In general, the degree of complication is a function of the excited determinants used to represent the wavefunction. The inclusion of triple and quadruple excited determinants still constitutes the best alternative in standard ab initio quantum chemistry. The methods MP4SDTQ, CCSD(T), and QCISDTQ, for instance, have the same scaling of N7, and their precision is approximately similar when basis sets containing f or higher angular momentum functions are used. Calculations with such a high degree of computational resources can obtain accuracies in relative energies approaching 1 kcal/mol. Calculations of this kind are limited for practical reasons to systems containing no more than 10 atoms due to their large use of computational resources. Calculations using more expensive basis sets using g and higher angular momentum functions have been reported with precisions approaching 0.1 kcal/mol. At present, calculations of this degree of precision are limited to systems containing only around 20 electrons.

The alternative method of solving the Schrödinger equation is density functional theory (DFT). This theory was developed simultaneously with the standard ab initio methods. However, it took a little longer to gain a very solid footing. The work of Hohenberg, Kohn, Sham, and Levy has contributed enormously to the recognition of DFT as an ab initio procedure. The theory and philosophy of DFT do not follow its standard ab initio counterparts, which delayed its acceptation as a formal theory. As demonstrated by the Hohenberg-Kohn theorem, the total energy of an electronic system can be expressed by a functional of the electron density. Actually, the implementation of several modern generalized gradient approximation (GGA) functionals attracted the quantum chemistry community to the use of DFT because of the great success of the GGAs in reproducing well-established experimental results that were not possible to obtain with the tools available in standard ab initio methods. The ab initio nature of GGA functionals such as those of Perdew and Wang, published in 1991, gave confidence to the systematic search for first principles functionals. The improvement of the functionals has been enormous, to the point that present calculations using a standard basis set including d-type functions and modern functionals are more precise than any of those using the sophisticated standard ab initio methods, including all those that scale as N7 and even in several cases where the latter are used with much larger basis sets. Most of the methods in DFT are based on the Kohn-Sham procedure, in which the Schrödinger equation is solved for the exact energy using a wavefunction from an ideal system of noninteracting electrons, bypassing the calculation of the real wavefunction, which is the major difficulty in the standard methods. Since the system of noninteracting electrons is constrained to possess the same electron density as the real interacting system, the quality of the energies is limited only by the functionals used to represent the total energy. The formal scaling of DFT methods is N3 or N4 depending on the particular method; however, as happens with the standard ab initio methods, in practice this scaling tends to be lower, about N2 for DFT, and work to reach even smaller scalings is in progress. The other great advantage of DFT methods lies in the fact that an...

Produkt-Details

• Erscheint lt. Verlag: 27.10.1998
• Mitarbeit: Herausgeber (Serie): Erkki J. Brandas, John R. Sabin, Michael C. Zerner; Chef-Herausgeber: Per-olov Lowdin
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Naturwissenschaften ► Biologie
• Themenwelt: Naturwissenschaften ► Chemie ► Analytische Chemie
• Themenwelt: Naturwissenschaften ► Chemie ► Physikalische Chemie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Angewandte Physik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Atom- / Kern- / Molekularphysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Festkörperphysik
• Themenwelt: Technik
• ISBN-10: 0-08-058258-3 / 0080582583
• ISBN-13: 978-0-08-058258-0 / 9780080582580