View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace at Waseda University Efficient Two-Component Relativistic Method to Obtain Electronic and Molecular Structures of Heavy-Element Systems ͷిࢠঢ়ଶ͓Αͼ෼ࢠߏ଄ͷͨΊͷܥॏݩૉ ޮ཰తͳ 2 ੒෼૬ର࿦๏ February 2017 Waseda University Graduate School of Advanced Science and Engineering Department of Chemistry and Biochemistry, Research on Electronic State Theory Yuya NAKAJIMA தౢɹ༟໵ Contents Chapter 1 General introduction 1 References . ................................ 6 Chapter 2 Theoretical background 7 2.1 Dirac Hamiltonian . ....................... 7 2.2 The IODKH method ....................... 9 2.3 The LUT scheme . ........................ 12 2.4 Spin-free and spin-dependent formalisms . ......... 14 2.5 The FCP method . ........................ 16 2.6 The GHF method . ........................ 19 2.7 The Analytical energy derivative for the GHF method . 21 References . ................................ 24 Chapter 3 Analytical energy gradient for spin-free infinite-order Douglas–Kroll–Hess method with local unitary transfor- mation 25 3.1 Introduction . .......................... 25 3.2 Theory and implementation . ................. 27 3.2.1 Energy gradient for IODKH . ........... 27 3.2.2 Analytical derivative for the space transformation matrices . ......................... 30 3.2.3 Energy gradient for LUT-IODKH . .......... 32 3.2.4 Implementation . ................... 33 i 3.3 Numerical assessments . .................. 34 3.3.1 Computational details . ................ 34 3.3.2 Numerical gradient values ............... 37 3.3.3 Accuracies of IODKH/C and LUT-IODKH/C methods 38 3.3.4 Computational cost of the LUT scheme ........ 45 3.3.5 Metal complexes . ................... 47 3.3.6 Heavier analogues of ethylene . ........... 48 3.3.7 Harmonic frequencies of diatomic molecules . 49 3.4 Conclusion . .......................... 50 References . ................................ 52 Chapter 4 Implementation of spin-dependent relativistic analytical energy gradient 57 4.1 Introduction . .......................... 57 4.2 Implementation . ........................ 60 4.3 Numerical assessments . .................. 61 4.3.1 Computational details . ................ 61 4.3.2 Accuracy of SD-IODKH and LUT-SD-IODKH . 63 4.3.3 Computational cost of (LUT-)SD-IODKH method . 67 − 4.3.4 Application in fac Ir(ppy3)............... 70 4.4 Conclusion . .......................... 72 References . ................................ 74 Chapter 5 Implementation of LUT-IODKH in GAMESS program 79 5.1 Introduction . .......................... 79 5.2 Implementation . ........................ 79 5.2.1 Relativistic correction in GAMESS program . 79 5.2.2 Threshold setting in the IOTC method ........ 80 ii 5.2.3 Integral evaluation at quadruple precision . 81 5.2.4 Combination with DC . ................ 82 5.2.5 Input options for LUT-IOTC . ........... 83 5.2.6 Major capabilities of LUT-IOTC . ........... 83 5.3 Numerical assessment ...................... 84 5.3.1 Computational details . ................ 84 5.3.2 Total energies of heavy atoms and molecules . 85 5.3.3 Threshold dependence of IOTC and LUT-IOTC . 86 5.3.4 Computational cost of LUT-IOTC . .......... 87 5.4 Conclusion . .......................... 89 References . ................................ 91 Chapter 6 Relaxation of core orbitals in the frozen core potential treat- ment 93 6.1 Introduction . .......................... 93 6.2 Theory and implementation . ................. 94 6.2.1 FCP with relaxation of core electrons . ........ 94 6.2.2 Implementation . ................... 96 6.3 Numerical assessments . .................. 98 6.3.1 Computational details . ................ 98 6.3.2 Computational cost of FCP-CR . ........... 99 6.3.3 Accuracy of FCP-CR . .................103 6.3.4 Core ionization energy and core level shift . 104 6.3.5 Accuracy of an iterative procedure between valence and core calculations . .................107 6.4 Conclusion . ..........................108 Appendix 6.A Dependence of core ionization potential energies 111 iii References . ................................113 Chapter 7 Relativistic effect on enthalpy of formation for transition metal complexes 117 7.1 Introduction . ..........................117 7.2 Computational details ......................119 7.3 Results and discussion ......................121 7.3.1 Accuracy of WFT . ...................121 7.3.2 Functional dependence . ................123 7.3.3 Geometry difference between PP and AE methods . 124 7.3.4 Effect of the levels of relativistic Hamiltonians . 124 7.3.5 Contribution of frozen core orbitals . .........126 7.4 Conclusion . ..........................128 References . ................................130 Chapter 8 General Conclusion 133 Acknowledgments 137 List of Achievements 139 iv List of abbreviations 2c two-component 4c four-component AE all-electron AO atomic orbital BSS Barysz-Sadlej-Snijders CA composite approach CCSD coupled cluster singles and doubles CCSD(T) coupled cluster singles, doubles, and perturbative triples CIE core ionization energy CLS core level shift CPU central processing unit DC divide-and-conquer DFT density functional theory DLU local approximation to the unitary decoupling transformation dmpe 1,2-bis(dimethylphosphino)ethane dOEI derivative of one-electron integral dppe 1,2-bis((pentafluorophenyl)phosphino)ethane dTEI derivative of two-electron integral ECP effective core potential ESC elimination of small components F-dppe 1,2-bis-((pentafluorophenyl)phosphino)ethane FCP frozen core potential v FCP-CR frozen-core potential with relaxation of core electrons FW Fouldy-Wouthuysen GAMESS General Atomic and Molecular Electronic Structure System GHF general Hartree–Fock HF Hartree–Fock HFR Hartree–Fock–Roothaan IODK infinite-order Douglas–Kroll–Hess IOTC infinite-order two-component LUT local unitary tranformation MAE mean absolute error MaxE maximum error MCP model core potential MP model potential MP2 second-order Møller–Plesset perturbation NESC normalized elimination of the small component NMR nuclear magnetic resonance NR non-relativistic OEI one-electron integral PP pseudo-potential ppy 2–phenylpyridine RA regular approximation RECP relativistic effective core potential RI resolution of identity SAC-CI symmetry-adapted cluster configuration interaction SCF self-consistent field SD spin-dependent SF spin-free vi TEI two-electron integral UT unitary transformation WFT wave function theory X2C exact two-component vii Chapter 1 General introduction Quantum chemistry plays an essential role in the qualitative and/or quantitative prediction and analysis of energetics and molecular properties such as geometries, spectra, and reactivities. To date, the basic, commonly used equation of quantum chemistry has been the NR Schrodinger¨ equation. In significant cases involving relativistic effects, such as orbital contractions and splittings in heavy-element systems, intersystem crossing, and core-electron related properties, relativistic effects are mostly accounted for through corrections to the NR treatment. On the other hand, using the Dirac equation, as a basic equation that satisfies the Lorentz invariance for electron motion, can account for relativistic effects. In 2002, Barysz et al. proposed a rigorous 2c method, referred to as the IODKH method1, that treats only electronic states. Seino and Nakai proposed the LUT technique that is able to perform efficient calculations without the loss of accuracy, using the original IODKH method2–6. The construction and extension of a theory based on both IODKH and LUT methods would enable the efficient treatment of relativistic effects using basic equations. Another efficient method for the treatment of heavy elements is the ECP method, which reduces the number of electrons that are treated explicitly7. The ECP method replaces the effect of core orbitals on the valence electrons with a potential. Constructing this potential to include relativistic effects provides a rel- ativistic treatment in a convenient manner. In 2014, Seino et al. proposed the FCP 1 method8, which describes core potentials by utilizing explicit core-orbital infor- mation obtained through atomic calculations. However, the effects of the chemical environment are not taken into account in the FCP method for molecules. This thesis extends the LUT-IODKH and FCP methods from a theoretical perspective, and applies the extended method to the determination of relativistic effects in significant systems. In order to perform geometry optimizations and frequency calculations, the analytical energy gradient for LUT-IODKH was developed and extended to the spin-dependent method9,10. This gradient method and the LUT scheme were then implemented in the GAMESS quantum chemical package11. Furthermore, FCP-CR, which relaxes the core orbitals that were treated as frozen orbitals by FCP, was also developed12. These methods were utilized for enthalpy of formation calculations, in the gas phase, of transition metal complexes. This thesis includes seven chapters, in addition this general introduction chapter (Chapter 1). Chapter 2 summarizes the theoretical background of the 2c relativistic method, as well as the IODKH, LUT, and FCP methods. Moreover, the energy and gradient expressions for the GHF method are provided. Chapter 3 extends the IODKH method to include analytical energy gradi- ents, and gradients combined with the LUT scheme. The energy gradient of the IODKH Hamiltonian, with respect to nuclear coordinates, is analytically derived. Numerical assessments