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Chapter 1 Scattering of X-rays and Neutrons 3 1.1 Outline of this chapter 3 1.2 Introduction to the theory of X-ray scattering 3 1.2.1 Classical treatment of X-ray scattering 3 1.2.2 Quantum-mechanical treatment: the first Born approximation 5 1.2.3 Scattering by a periodic crystal 7 1.2.4 The structure factor formalism in terms of atomic densities 9 1.3 Resonance scattering of X-rays 11 1.3.1 Classical treatment 11 1.3.2-Quantum-mechanical treatment: the second Born approximation 13 1.3.3 The power series expansion of the scattering operator 15 1.3.4 The optical theorem and the relation between /" and /' 16 1.4 Neutron scattering 18 1.4.1 Properties of neutrons 18 1.4.2 The neutron scattering length 19 Chapter 2 The Effect of Thermal Vibrations on the Intensities of the Diffracted Beams 22 2.1 The normal modes of a crystal 23 2.1.1 Phonons, internal and external modes 23 2.1.2 The frequency of the normal modes 23 2.2 The effect of thermal vibrations on the Bragg intensities 27 2.2.1 The Born-Oppenheimer approximation 27 2.2.2 The harmonic temperature factor 28 viii Contents 2.2.3 Beyond the harmonic approximation 31 2.2.4 Comparison of the anharmonic formalisms 36 2.2.5 Quantum-statistical treatments 37 2.3 The relation between the atomic temperature factors and lattice dynamics 40 2.3.1 General expression 40 2.3.2 The Debye approximation 41 2.3.3 The rigid-body model for molecular crystals 42 2.3.4 The rigid-bond test 48 Chapter 3 Chemical Bonding and the X-ray Scattering Formalism 49 3.1 The breakdown of the independent-atom model 49 3.1.1 Qualitative considerations 49 3.1.2 The electron density and the LCAO formalism 51 3.2 Improved scattering models 55 3.2.1 The spherical atom kappa formalism 55 3.2.2 Modified spherical scattering factor for the hydrogen atom 56 3.2.3 Examples of results obtained with the K-formalism 57 3.2.4 The multipole description of the charge density of aspherical atoms 59 3.2.5 Aspherical atom scattering factors 67 3.2.6 The aspherical density functions of the Hirshfeld formalism 70 Chapter 4 Least-Squares Methods and Their Use in Charge Density Analysis 72 4.1 Least-squares equations 72 4.1.1 Background 72 4.1.2 General formalisms for linear least-squares 72 4.1.3 Explicit expressions for structure factor least-squares 74 4. L4 Variances and covariances of the least-squares parameter estimates 76 4.1.5 Uncorrelated linear combinations of variables 79 4.2 The least-squares parameters in charge density analysis 79 4.2.1 The parameters in a charge density refinement 79 4.2.2 Parameter restrictions imposed by site and local symmetry and chemical equivalence 80 4.2.3 The scale factor 81 4.3 Physical constraints of the electron density 83 4.3.1 The electroneutrality constraint 83 4.3.2 The Hellmann-Feynman constraint 85 4.4 Joint refinement of X-ray and neutron data 86 4.4.1 The use of complementary information 86 4.4.2 Differences in temperature parameters 86 4.4.3 Relative weighting of the X-ray and neutron data 87 4.4.4 Estimate of the goodness of fit 88 Contents xi 9.2.1 The Hohenberg-...

The molecular structure and dynamics of the photoexcited metal-to-ligand-charge-transfer (MLCT) state of [Cu(I)(dmp)(2)](+), where dmp is 2,9-dimethyl-1,10-phenanthroline, in acetonitrile have been investigated by time-domain pump-probe X-ray absorption spectroscopy, femtosecond optical transient spectroscopy, and density functional theory (DFT). The time resolution for the excited state structural determination was 100 ps, provided by single X-ray pulses from a third generation synchrotron source. The copper ion in the thermally equilibrated MLCT state has the same oxidation state as the corresponding copper(II) complex in the ground state and was found to be penta-coordinate with an average nearest neighbor Cu-N distance 0.04 A shorter than that of the ground state [Cu(I)(dmp)(2)](+). The results confirm the previously proposed "exciplex" structure of the MLCT state in Lewis basic solvents. The evolution from the photoexcited Franck-Condon MLCT state to the thermally equilibrated MLCT state was followed by femtosecond optical transient spectroscopy, revealing three time constants of 500-700 fs, 10-20 ps, and 1.6-1.7 ns, likely related to the kinetics for the formation of the triplet MLCT state, structural relaxation, and the MLCT excited-state decay to the ground state, respectively. DFT calculations are used to interpret the spectral shift on structural relaxation and to predict the geometries of the ground state, the tetracoordinate excited state, and the exciplex. The DFT calculations also indicate that the amount of charge transferred from copper to the dmp ligand upon photoexcitation is similar to the charge difference at the copper center between the ground-state copper(I) and copper(II) complexes.