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Mayer, Istvan

Simple Theorems, Proofs, and Derivations in Quantum Chemistry (Kartoniert / Broschiert / Paperback)

Springer US, Dezember 2010

Softcover reprint of the original 1st ed. 2003 - 352 S. - Sprache: Englisch - 254x177x22 mm Previously published in hardcover

ISBN: 1441933891 EAN: 9781441933898

Since 1983 I have been delivering lectures at Budapest University that are mainly attended by chemistry students who have already studied quantum chem­ istry in the amount required by the (undergraduate) chemistry curriculum of the University, and wish to acquire deeper insight in the field, possibly in prepara­ tion of a master's or Ph.D. thesis in theoretical chemistry. In such a situation, I have the freedom to discuss, in detail, a limited number of topics which I feel are important for one reason or another. The exact coverage may vary from year to year, but I usually concentrate on the general principles and theorems and other basic theoretical results which I foresee will retain their importance despite the rapid development of quantum chemistry. I commonly organize my lectures by treating the subject from the begin­ ning, without referring explicitly to any actual previous knowledge in quantum chemistry-only some familiarity with its goals, approaches and, to a lesser ex­ tent, techniques is supposed. I concentrate on the formulae and their derivation, assuming the audience essentially understands the reasons for deriving these results. This book is basically derived from the material of my lectures. The spe­ cial feature, distinguishing it from most other textbooks, is that all results are explicitly proved or derived, and the derivations are presented completely, step by step. True understanding of a theoretical result can be achieved only if one has gone through its derivation.

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I. The Born-Oppenheimer Hamiltonian. 1. Separating the center of mass motion in quantum mechanics. 1.1. Reducing the two-body problem to two one-body ones. 1.2. The center of mass in quantum mechanics 1.3. Free atoms and atomcules. 2. The Born--Oppenheimer approximation. 2.1. Introductory remarks. 2.2. The Born-Oppenheimer separation. 2.3. Why the Born-Oppenheimer separation is not exact? 2.4. Approximate decoupling. 2.5. A note on the Born-Oppenheimer separation. II. General Theorems And Principles. 1. The variation principle. 1.1. The Rayleigh quotient. 1.2. The variation principle for the ground state. 1.3. The variation principle as an equivalent of the Schrodinger equation: a useful formulation of the variation principle. 1.4. Eckart's inequality. 1.5. Excited states. 2. The Hellmann - Feynman theorem. 2.1. The differential Hellmann - Feynman theorem. 2.2. The integral Hellmann -Feynman theorem. 3. The virial theorem in quantum mechanics. 3.1. Time dependence of a physical quantity. 3.2. The virial theorem. 3.3. Scaling - a connection with the variation principle. 3.4. The virial theorem in the Born-Oppenheimer approximation. 3.5. The virial theorem and the chemical bonding. III. The Linear Variational Method And Lowdin's Orthogonalization Schemes. 1. The linear variational method (Ritz -method) 2. Lowdin's symmetric orthogonalization. 2.1. Matrix SAND-1/2. 2.2. The S∧-1/2 transformation. 2.3. The Lowdin basis. 2.4. The stationary property of Lowdin's symmetric orthogonalization scheme. 2.5. Lowdin-orthogonalization: a two-dimensional example. 3. Linear independence of the basis and Lowdin's canonic orthogonalization. 3.1. Eigenvalues of the overlap matrix: a measure for the linear. 3.2. Lowdin's canonic orthogonalization. IV. Perturbational Methods. 1. Non-degenerate Rayleigh-Schrodinger perturbation theory. 1.1. The problem. 1.2. `Algebraic' expansion. 1.3. The use of the reduced resolvent in the Rayleigh-Schrodinger perturbation theory. 1.4. Wigner's 2n+1 theorem. 2. Variational-perturbational method: the Hylleraas-functional.3. Degenerate Rayleigh-Schrodinger perturbation theory. 4. Brillouin-Wigner perturbation theory. 4.1. The size-consistency problem. 5. Size consistency of the Rayleigh-Schrodinger perturbation. 5.1. Formal considerations based on the properties of power series. 5.2. Size consistency of the perturbational expansions. 6. Lowdin's partitioning method. V. Determinant Wave Functions. 1. Spin-orbitals. 2. Many-electron spin states. 3. Slater determinants. 3.1. Two-electron examples. 4. The antisymmetrizing operator. 4.1. The projection character of the antisymmetrizing operator. 4.2. Commutation properties of the antisymmetrizing operator. 5. Invariance of the determinant wave function with respect of. 6. Matrix elements between determinant wave functions. 6.1. Overlap. 6.2. One-electron operators. 6.3. Two-electron operators. 7. Lowdin's pairing theorem and its generalization. 7.1. Lowdin's pairing theorem. 7.2. Karadakov's extended pairing theorem. 8. Theorem about the existence of orbitals of special structure. 8.1. Existence theorem. 8.2. A posteriori determination. VI. The Hartree-Fock Method. 1. The variation principle for single determinant wave functions: the Brillouin theorem. 1.1. The Brillouin theorem for the determinant with the absolute minimum of the energy. 1.2. The Brillouin theorem for a determinant with a stationary energy. 1.3. Algorithm of calculations based on the Brillouin theorem. 2. The Hartree-Fock equations. 2.1. Unrestricted Hartree-Fock equations. 2.2. Alternative derivation by using Lagrangian multipliers. 2.3. Alternative derivation by using specified variations. 3. Koopmans theorem. 3.1. The orbital energies and the total energy. 4. The RHF method. 4.1. The RHF and UHF schemes 4.2. Symmetry and the RHF method. 4.3. The "dissociation catastrophe"; different Hartree-Fock methods. 4.4. Singlet and triplet excitations. 5. Finite basis Hartree-Fock theory. 5.1. Hartree-Fock-Roothaan equations. 5.2. The P-matrix. 5.3. Example of using projectors in the LCAO framework: the UHF equations for successive optimization of the orbitals. 5.4. The Fock matrix and the energy.6. Restricted open shell Hartree-Fock method. 6.1. The ROHF equations. 6.2. The coupling coefficients for some open shell systems. 7. The gradient of the energy. VII. Population Analysis, Bond Orders And Valences. 1. Mulliken's population analysis. 1.1. Electron density. 1.2. Population analysis. 2. Bond orders and valences. 2.1. Bond order index. 2.2. Valence indices. 2.3. Exchange density and bond order. 2.4. Bond orders in three-center bonds. VIII. The Electron Correlation. 1. The CI expansion. 2. The correlation energy and Nesbet's theorem. 3. Methods of electron correlation calculations. 3.1. The CI method. 3.2. MC-SCF methods. 3.3. The MBPT or Møller-Plesset method. 3.4. The Coupled Cluster Approach. IX. Miscellaneous. 1. The 1/Z expansion. 2. The cusp condition. 2.1. The nuclear cusp. 2.2. The electronic cusp. 3. The asymptotical behaviour of the wave function at very large. 4. Basis functions: the Gaussian product theorem. 5. The integral transformation problem. Appendices.

ISBN 1-441933-89-1, ISBN 1-44-193389-1, ISBN 1-44193-389-1, ISBN 1-441-93389-1, ISBN 1-4419-3389-1

ISBN 978-1-441933-89-8, ISBN 978-1-44-193389-8, ISBN 978-1-44193-389-8, ISBN 978-1-441-93389-8, ISBN 978-1-4419-3389-8

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