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UC Berkeley UC Berkeley Electronic Theses and Dissertations UC Berkeley UC Berkeley Electronic Theses and Dissertations Title The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics Permalink https://escholarship.org/uc/item/82w6491p Author Stuck, David Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics by David Edward St¨uck A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Martin Head-Gordon, Chair Professor William H. Miller Professor Alexis T. Bell Summer 2015 The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics Copyright 2015 by David Edward St¨uck 1 Abstract The Importance of Zeroth-Order Approximations in Molecular Quantum Mechanics by David Edward St¨uck Doctor of Philosophy in Chemistry University of California, Berkeley Professor Martin Head-Gordon, Chair The work herein is concerned with developing computational models to understand molecules. The underlying theme of this research is the reassessment of zeroth-order approximations for higher-level methods. For second-order Møller-Plesset theory (MP2), qualitative failures of the Hartree-Fock orbitals in the form of spin contamination can lead to catastrophic errors in the second order energies. By working with orbitals optimized in the presence of correla- tions, orbital-optimized MP2 can fix the spin contamination problem that plague radicals, aromatics, and transition metal complexes. In path integral Monte Carlo for vibrational energies, the zeroth-order propagator is typically chosen to be the most general possible, the free particle propagator; we chose to be informed by the molecular structure we have already attained and apply a propagator based on the harmonic modes of the molecule, improving sampling efficiency and our Trotter approximation. i Contents Contents i List of Figures iii List of Tables vi 1 Introduction 1 1.1 Background . 1 1.2 Electron Correlation . 5 1.3 Statistical Quantum Thermodynamics . 8 1.4 Outline . 9 1.5 Additional Work . 11 2 On the Nature of Electron Correlation in C60 13 2.1 Introduction . 13 2.2 Results and Discussion . 15 2.3 Conclusion . 21 3 Regularized Orbital-Optimized MP2 22 3.1 Introduction . 22 3.2 Theory . 26 3.3 Results and Discussion . 28 3.4 Conclusion . 34 4 Stability Analysis without Analytical Hessians 35 4.1 Abstract . 35 4.2 Introduction . 35 4.3 Method . 37 4.4 Results . 39 4.5 Conclusions . 44 4.6 Acknowledgements . 45 ii 5 Exponential Regularized OOMP2 for Dissociations 46 5.1 Introduction . 46 5.2 Theory . 47 5.3 Results . 48 5.4 Conclusion . 50 5.5 Acknowledgements . 51 6 Regularized CC2 53 6.1 Introduction . 53 6.2 Computational Methods . 55 6.3 Results and Discussion . 56 6.4 Conclusion . 58 6.5 Acknowledgements . 59 7 Path Integrals for Anharmonic Vibrational Energy 60 7.1 Introduction . 60 7.2 Theory . 61 7.3 Results and Discussion . 65 7.4 Conclusion . 70 7.5 Acknowledgements . 71 References 74 iii List of Figures 2.1 Natural orbital occupation numbers of UHF spincontaminated singlets for C36 i i and C60. Orbitals are numbered as a fraction of the total π space (i.e. 36 or 60 th for the i π orbital of C36 or C60 respectively). 17 2.2 Unpaired electron density of singlet (top) and triplet (bottom) C60 (left) and C36 (right) plotted at isovalue 0.006 A˚−3, with shading determined by the sign of the spin density as described in the text. 18 2.3 Natural orbital occupation numbers from O2 calculations on singlet C36 and C60. Orbitals are numbered as a fraction of the total π space. 20 3.1 Li2 dissociation curve for MP2 using restricted and unrestricted orbitals and for OOMP2 with a cc-pVDZ basis. RMP2 dissociates incorrectly and UMP2 distorts the equilibrium description while OOMP2 gets the best of both worlds by continuously connecting the two regimes, albeit with a kink due to a slight discontinuous change to the orbitals upon unrestriction. 24 3.2 Dependence of the OOMP2 energy (the standard RIMP2 energy without singles contribution) on the two occupied-virtual mixing angles for the hydrogen molecule in the STO-3G basis at 0.74 A.˚ The region around the RHF minimum at (0◦; 0◦) is well behaved. 29 3.3 Dependence of the OOMP2 energy on the two occupied-virtual mixing angles for the hydrogen molecule in the STO-3G basis at 4.0 A.˚ Divergences appear for orbitals with unfavorable HF energies but very large negative MP2 energy due to HOMO-LUMO energy coalescence. There is a stable minimum near the UHF solution around (140◦; 40◦), but it is not the global minimum due to the divergences. 29 3.4 δ-OOMP2 orbital energy surface with level shifts, δ, of 100 mEh (left) and 400 mEh (right) for the hydrogen molecule in the STO-3G basis at 4.0 A.˚ The level shift of 400 mEh has restored the solution near the UHF orbitals to be the global minimum and has removed the divergences. 30 3.5 RMS error on the G2 test set of atomization energies for δ-OOMP2, δ-RIMP2, and correlation scaled RIMP2 and OOMP2 as a function of the regularization (2) parameter δ (bottom) or scaling parameter, s, given by Es = E0 + sE (top). 31 iv 3.6 RMS errors of δ-OOMP2 relative to standard RIMP2 on various test sets. With- out regularization OOMP2 performs worse than RIMP2 for the G2 and S22 test sets but a level shift of 400 mEh improves δ-OOMP2 over RIMP2 and unregular- ized OOMP2 for all test sets. 33 3.7 (a) Bond length errors vs. CCSD(T) of OOMP2, δ-OOMP2, and MP2 for five small radicals. (b) Harmonic frequencies plotted against CCSD(T) for the same five radicals. R2 values for frequencies are 0.979, 0.998, and -0.003 for OOMP2, δ-OOMP2, and MP2 respectively. MP2 and reference CCSD(T) values taken from the work of Bozkaya[100]. 33 4.1 Potential curves (green for unrestricted and red for restricted, where it differs from unrestricted) for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix (purple for internal stability of the unrestricted solution, blue for external stability of the restricted solution, where it differs from unrestricted) at the Hartree-Fock (HF) level. The lowest energy solution changes character from restricted to unrestricted when the former becomes unstable. 40 4.2 Potential curves for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix using orbital-optimized MP2 (OOMP2) in the cc-pVDZ basis. The format follows Figure 6.1. OOMP2 behaves qualitatively differently from HF (see Figure 4.1). The restricted solution is stable (positive eigenvalue) to spin-polarization at all bond-lengths, and a distinct stable unrestricted solution appears at partially stretched bondlengths. 41 4.3 The dependence of the OOMP2 energy of H2 in a minimal basis on the spin polarization angle (see text for definition) at a series of bond-lengths around the critical value at which the character of the lowest energy solution changes. There are two local minima, one restricted and one unrestricted, at these bond-lengths, and at the critical bond-length the nature of the lowest energy solution switches discontinuously. 42 4.4 Potential curves for the dissociation of H2 and the associated lowest eigenvalues of the stability matrix using regularized orbital optimized MP2 (δ-OOMP2) in the cc-pVDZ basis. The format follows Figure 6.1. δ-OOMP2 behaves qualitatively differently from OOMP2 (see Figure 4.2), but is similar to HF (see Figure 4.1). The restricted solution becomes unstable at a critical bond-length, beyond which the unrestricted solution is lowest in energy. 43 4.5 The dependence of the δ-OOMP2 energy of H2 in a minimal basis on the spin polarization angle (see text for definition) at a series of bond-lengths around the critical value at which the character of the lowest energy solution changes. For any given bond-length there is only one local minimum, which changes character from restricted to unrestricted at the critical bond-length. 44 v 5.1 Dissociation curve of ethane in an aug-cc-pVTZ basis. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals. 48 5.2 Dissociation curve of ethene in an aug-cc-pVTZ basis. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals. 49 5.3 Dissociation curve of ethane in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals. 50 5.4 Dissociation curve of ethene in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals. 51 5.5 Dissociation curve of ethyne in an aug-cc-pVTZ basis for σ-OOMP2 with an σ value of 3.2. hS2i of the unrestricted solution and lowest Hessian eigenvalue for the restricted solutions plotted to show discontinuity in orbitals. 52 6.1 δ-CC2 RMSE for various ground state test sets divided by RIMP2 RMSE on the same sets for various values of δ. .......................... 56 6.2 Ozone symmetric dissociation curve at angle 142.76◦ for CC2 with regularization parameters 0, 100, 150, and 200 mEh and CCSD in an aug-cc-pVTZ basis.
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