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Copper Dimer Theoretical Studies CALIFORNIA STATE UNIVERSITY, NORTHRIDGE Theoretical Studies of Copper Clusters A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Chemistry by Courtney Marie Sams August 2013 The thesis of Courtney Marie Sams is approved: _______________________________ ______________ Dr. Daniel B. Curtis, Ph.D. Date _______________________________ ______________ Dr. Simon J. Garrett, Ph.D. Date _______________________________ ______________ Dr. Jussi M. Eloranta, Ph.D., Chair Date California State University, Northridge ii Dedication This thesis is dedicated to the four people that are not here now but meant so much to me in my life: Jean Marie Fractious for giving me tenacity, Selma Sams for teaching me that being yourself is primary, Gregory Aaron Fractious for teaching me to be a good friend to all and Carol (Pat) Graham for always laughing at my randomness even when no one else would. iii Acknowledgements I’d like to thank everyone who has helped me along on this journey: Thank you Mom for passing on the IQ so I can (mostly) understand my research. Thank you Dad for introducing me to the nerdiness which is so fundamentally me. Jared (MB) for his amazing unflagging support even when he did not understand my ‘Sciency/Rain-man’ talk. My numerous aunts and uncles who are there for me and my many cousins who knew me when all I wanted to do was read! Thank you to my friends who stand by me and encourage me even when I am ready to give up and pursue other paths of life (Mojghan, Anna, Cassandra and of course, always Martha). Thanks to the Department of Chemistry and Biochemistry staff (Dr. Kelson, Irene McGee and Migdonia (Sonia) Martinez) and the many other professors that had to put up with my method of expression. Lastly, but not least, I’d like to thank my research advisor, Dr. Jussi Eloranta, for putting up with my need to work to pay for school in order to continue the research that I love. iv Table of Contents Signature Page ii Dedication iii Acknowledgements iv List of Tables vi List of Figures vii Abstract viii 1: Introduction to Computational Chemistry 1.1 Schrödinger Equation (time independent) 1 1.2 Born-Oppenheimer Approximation 3 1.3 Hartree-Fock 4 1.4 Coupled Clusters Method 8 1.5 Equation-of-Motion Coupled Clusters Method 10 1.6 Gaussian Basis Sets 10 1.7 Basis Set Superposition Error 12 1.8 Density Functional Theory 13 1.9 Accuracy of DFT Functionals 19 1.10 Relativistic Effects 20 2: Electronic structure of homonuclear diatomic molecules 2.1 Molecular orbitals 21 2.2 Optical Spectroscopy 26 3: Density Functional Theory Calculations of small copper clusters 3.1 Copper Dimers and Helium Impurities (CC-EoM) 33 3.2 Copper Clusters (DFT) 45 References 51 Appendix A: Journal of Physical Chemistry A, Vol. 115, pg. 1-34 (2011) 58 v List of Tables Table 1: Comparison of performance of DFT methods by mean 19 absolute deviations Table 2: Comparison of the performance of DFT methods with 20 respect to G2 Table 3: Electronic origins for ground and excited states of Cu2 40 Table 4: Equilibrium bond lengths of Cu2 (Å) 40 -1 Table 5: Harmonic frequencies for Cu2 (cm ) 41 Table 6: Parametrized potentials fitted to the form �(�) = 41 �� − �� − �� , in Å-cm units -1 Table 7: Vibrational frequencies of Cu2 (cm ) 47 Table 8: DFT Optimized Copper - Copper Bond Lengths (Å) 48 Table 9: Binding energy He – Cu2 (meV) 49 vi List of Figures Figure 1: Guide used to determine the order of atomic orbitals in 22 the overall molecular orbital diagram Figure 2: Molecular orbital diagram for the F2 molecule 24 Figure 3: Visualization of bonding and antibonding molecular 25 orbitals Figure 4: Molecular Orbital diagram for the excited F2 molecule 25 Figure 5: Flow chart utilized in point group determination 29 Figure 6: Direct Product table for the C3v point group 30 Figure 7: Images of thermomechanical He fountain 34 Figure 8: Thermomechanical He fountain schematic 35 Figure 9: LIF imaging of ablated Cu2 36 Figure 10: Visualization of copper dimer – helium interaction 39 Figure 11: Detailed view of potential energy curves of ground 42 state (X) and the six excited states for Cu2 Figure 12: Potential energy curve comparing ‘T’ and ‘L’ 44 configurations for Cu2 – He Figure 13: Optimized copper cluster geometry 48 Figure 14: Visualization of Copper – Helium interaction 49 vii Abstract Theoretical Studies of Copper Clusters A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Chemistry by Courtney Marie Sams Master of Science in Chemistry The ground and several excited states of copper dimer (X, A, B and C) were calculated by the coupled clusters equation-of-motion method. A qualitative agreement with gas phase spectroscopic observables, such as the electronic origins, harmonic and anharmonic vibrational frequencies and radiative lifetimes, were obtained. For the B and C states the calculations verify the experimentally generated potential energy curves for these states and further identify the weakly allowed A state to originate from the spin-orbit mixture of the B and A states. It was found that the present level of accuracy could only be reached when scalar relativistic corrections (Douglas-Kroll Hamiltonian; DKH) were included along with the DKH compatible triple zeta level basis set. Finally, the interaction between the copper dimer and atomic helium was calculated to elucidate the possible solvation structure of the dimer in superfluid helium. It was found that while Cu atoms are heliophobic, Cu2 dimers are heliophilic and should remain inside helium droplets. Using a pseudopotential method in the generalized gradient approximation and plane wave expansion, the binding energy and equilibrium bond lengths were calculated for Cu clusters from 2 – 4 and 7 atoms. viii ix Chapter 1: Introduction to Computational Chemistry 1.1 Time Independent Schrödinger Equation Erwin Schrödinger, an Austrian physicist and theoretical biologist, in 1926 proposed the wave equation as the foundation of quantum mechanics (Schrödinger’s equation): (1) � = E where � is the Hamiltonian operator that represents total energy and Ei’s are the numerical values representing energy. Wavefunctions, i’s, contain all the general experimentally determinable information of the system. Eq. (1) has many possible eigenvalue and eigenfunction pairs that satisfy it. Mathematically Eq. (1) is called an eigenvalue problem. According to Bohr’s interpretation, ||2 is the probability density that when integrated over all space is normalized to unity (∫ ||dτ = 1). The Hamiltonian operator in Eq. (1) includes the potential energy of the electron-electron repulsion, the nuclear-electron attraction and the nuclear – nuclear repulsion and the kinetic energy terms: ℏ ∆ ℏ (2) � = − ∑ − ∑ ∆ − ∑ ∑ + ∑ ∑ + ∑ ∑ where the first and second terms are the kinetic energy of the nuclei and the electrons, respectively, the third term is the electron-nuclear attraction, and the fourth and fifth terms are the repulsions that occur between any electrons or any nuclei. N is the number of nuclei in the system, n corresponds to the number of electrons, the reduced Planck’s 1 -34 -31 constant is ℏ, equal to 1.054 x 10 J∙s, me is the mass of the electron (9.109 x 10 kg), ∆ (≡ � ) is the Laplacian operator that acts on the electron i coordinates, ZI is the -19 atomic number for the Ith nuclei, e is the elementary charge, 1.602 x 10 C, and εo is the vacuum permittivity, which is equal to 8.854 x 10-12 F∙m-1. It is currently not known how to solve the Schrödinger equation analytically for any system with more than one electron. As a result, approximations must be made. Two such approximations are the Variational Principle and the Born-Oppenheimer (BO) approximation, discussed in Section 1.2. Although very different in nature, both have been used to help solve the Schrödinger equation for molecular systems. The Variational Principle approximates the ground state of a system. ∗ ∫ (3) Ε = ∗ ∫ Typical application of the Variational Principle would be the following: 1. Choose � 2. Calculate Eref using � and Eq. (3) 3. Choose � 4. Calculate E using � and Eq. (3) 5. If E < Eref, {� = � and Eref = E}, Repeat step 3 if |E - Eref| > ε where ε is some predetermined energy threshold. Ending this process is difficult because the actual Eo is not known. According to the Variational Principle, the trial energy, Etrial is found to be greater than or equal to the minimum energy, Eo (Etrial ≥ Eo). The Hamiltonian operator, �, in Eq. (1) is hermitian which means that all eigenvalues and therefore also energies are real. Furthermore all eigenfunctions can be made orthogonal ∗ (∫ ψ ψ dτ = 0, when � ≠ �), and normalized (orthonormal). The eigenfunctions of Eq. 2 (1) form a complete basis set. A complete basis set means that it is possible to represent any function through a linear combination of these basis functions. A more detailed concept of basis function, will be discussed in Section 1.5. 1.2 Born-Oppenheimer Approximation Solving Eq. (1) for a non – hydrogen-like molecule can be computationally expensive and extremely difficult if certain approximations are not utilized. One of these approximations is the Born-Oppenheimer (BO) approximation. The BO approximation separates the molecular wave function into the nuclear and electronic parts by assuming that the nuclei are fixed in place and only the electronic degrees of freedom need to be considered.1 The premise of this approximation is that the nucleus is so much heavier (1800 times greater) than an electron and it can therefore be assumed to be stationary while the electrons orbit around the nucleus. In the BO approximation, the total wavefunction is written as a product of the nuclear and electronic parts (4) Ψ(R, r) = Ψ(R) ∙ Ψ(R, r) where Ψtotal is the total wavefunction, Ψel is the electronic wavefunction, Ψn is the nuclear wavefunction, R corresponds to the nuclei and r to the electrons.
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