Relativistic Calculations of NMR Shifts Prepared by: Ben Joseph R. Cuyacot 2 Why Relativistic Quantum Chemistry? (Because the periodic table doesn’t end with Kr ) ➒ Recall the non-relativistic Hamiltonian ෑ𝑯 𝒏𝒓𝒆𝒍 = ෍ π’Š=𝟏 ΰ·π’‘π’Š 𝟐 πŸπ’Ž + ෍ π’Š 𝑽 π‘΅π’Š + 𝑽 𝒆𝒆 + 𝑽𝒏 𝒖𝒄 ෝ𝒑 π’Š 𝟐 πŸπ’Ž is the non-relativistic kinetic energy operator ➒ In heavy elements the inner shell electrons move with a speed comparable with speed of light. ➒ So the core electrons of heavy atoms show sizable relativistic effect. For accurate predictions of various properties of chemical system containing heavy elements. π’Ž 𝒓𝒆𝒍 = π’Ž 𝒐 𝟏 βˆ’ 𝒗 𝟐/𝒄 𝟐 ➒ Properties like NMR shift is very sensitive to this effect. 3 Relativistic Quantum Theory? ➒ The marriage of the two greatest concepts of Physics of the 20th century, the Quantum Theory and the Theory of Relativity. ➒ Dirac labor alone on the problem for 3 years before publishing it in 1928, The Quantum Theory of the Electron (https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1928.0023) ➒ If we compare S.E. Hamiltonian from the Dirac Equation, we can see that there’s an Ξ± and Ξ² terms, unknown terms in which Dirac introduced to proceed with his labor. SchrΣ§dinger Equation Dirac Equation ෑ𝐻 = βˆ’ 1 2 βˆ‡2 + 𝑣(π‘Ÿ) ෑ𝐻 = 𝛼 βˆ™ 𝑝 + 𝛽𝑐2 + 𝑣(π‘Ÿ) ➒ These supposed introduction of Ξ± and Ξ² terms (4 X 4 matrices) naturally suggest its four component nature as well as positive and negative energy solutions. 4 Relativistic Quantum Chemistry ➒ Why Relativistic effects matter 𝐻 𝑅 = 𝐻 𝑁𝑅 βˆ’ 𝑝2 8𝑐2 + βˆ†π‘£ 8𝑐2 + 1 2𝑐2 1 π‘Ÿ 𝑑𝑣 π‘‘π‘Ÿ 1 βˆ™ 𝑠 mass Darwin spin-orbit -velocity βˆ’ 𝑝2 8𝑐2 mass-velocity: variation of the mass with the velocity ( affects contraction and stabilization of sand p shells; expansion and destabilization of the d and f-shells) + βˆ†π‘£ 8𝑐2 Darwin: correction to the non-local interaction between electron and Coulomb field + 1 2𝑐2 1 π‘Ÿ 𝑑𝑣 π‘‘π‘Ÿ 1 βˆ™ 𝑠 spin-orbit coupling: interaction of the electron spin magnetic moment with magnetic field due to the motion of the (affects the character of orbitals). These part accurately describes with the β€œfine structure” split of the Hydrogen spectrum, in Michelson-Morley (1887) 5 Two-Component Relativistic Methods ➒ High computational cost of four-components relativistic calculations has motivated the development of computationally less demanding two-components Hamiltonians. ➒ Two component relativistic Hamiltonians (involving only positive energy orbitals): pseudopotential and allelectron methods. ZORA: accurate and efficient relativistic DFT ➒ The zeroth order regular approximation (ZORA) to the Dirac equation accurately and efficiently treats relativistic effects in chemistry. ZORA can be applied with spin-orbit coupling or as scalar correction only. β€’ Spin Orbit Coupling can be included self consistently β€’ All electron relativistic basis sets for all elements β€’ Available for most spectroscopic properties β€’ Include relativistic effects on structure and reactivity From Dirac to ZORA equation ➒ From Four-component Dirac (Fock Matrix) 𝑉𝑛𝑒𝑐 + 𝐽φφ βˆ’ 𝐾φφ + 𝐽π‘₯π‘₯ π‘πœŽ βˆ™ 𝑝 βˆ’ π‘˜Ο†π‘₯) π‘πœŽ βˆ™ 𝑝 βˆ’ π‘˜π‘₯Ο† βˆ’2𝑐2 + 𝑉𝑛𝑒𝑐 + 𝐽φφ βˆ’ 𝐾φφ + 𝐽π‘₯π‘₯ βˆ™ πœ‘ Ο‡ = 𝐸 πœ‘ Ο‡ Ξ¨= πœ‘ Ο‡ ➒ To two-components zeroth order regular Approximation 𝐻 𝑍𝑂𝑅𝐴 = 𝑉 + 𝒑 𝒄 𝟐 πŸπ’„ πŸβˆ’π‘½ 𝒑 + 𝒄 𝟐 (πŸπ’„ πŸβˆ’π‘½) 𝟐 𝝈 βˆ™ (πœ΅π‘½ Γ— 𝒑) ➒ The approximation is very convenient as we can now isolate the scalar and spin-orbit effects from the results of the calculation. 6 From Dirac to ZORA After a long derivation 𝐻 𝑍𝑂𝑅𝐴 Ξ¨ 𝑍𝑂𝑅𝐴 = 𝐸 𝑍𝑂𝑅𝐴 Ξ¨ 𝑍𝑂𝑅𝐴 7 ➒ ReSpect (RelativisticSpectroscopy) program enables the prediction and understanding of molecular and material properties using full Dirac Hamiltonian. (http://www.respectprogram.org/ ) ➒ ADF (Amsterdam Density Functional) program enables the prediction and understanding of molecular and material properties using Zeroth Order Approximation (ZORA) (https://www.scm.com/product/adf/) ReSpect vs ADF 8 ➒ Calculate the NMR properties of hydrogen in HI, HCl, and Benzene ➒ Use the prepared input files distributed in IS ( HCl_Scalar.inp, nmr.inp, etc) ➒ Take a look at the script run_adf.sh to see how adf run works. NOTE: There are two parts of adf calculation, SCF module-Calculation and the calculation of nmr shielding. ➒ After the successful NMR run. Check the .out file, Look for the keyword β€œ PARAMAGNETIC (), DIAMAGNETIC () and SPIN ORBIT (NMR SHIELDING TENSOR)” and below it is the β€œisotropic” shielding value(s) of the atom(s) sought. NOTE: PARA eigenvalues associated with an excited and the ground electronic state, DIA contributions are associated from the ground-state. (this is beyond the scope of our session) Exercise (ADF) Recall: The NMR Shifts [𝜹] of specific atom (π’Š) in a molecule can be obtained subtracting NMR shielding [𝝈] from its reference πœΉπ’Š = 𝝈 𝒓𝒆𝒇 βˆ’ πˆπ’Š STEP ADF Calc 9 ➒ After the successful of ZORA Scalar and ZORA Spin-Orbit Approximations, compare some the values with the experiment, you should be able to complete this table: Calc. Shielding Calc. Shielding Calc. Shielding Calc Shift Exp Shift (ref-TMS) Total Οƒref para dia Οƒi (para + dia) Ξ΄i (Οƒref - Οƒi) SO Ξ΄i + SO *Ξ΄exp HCl_Scalar 31.35 0.45 HCl_SO 31.41 0.45 HI_Scalar 31.35 13.25 HI_SO 31.41 13.25 Benzene_Scalar 31.35 7.15 Benzene_SO 31.41 7.15 10 ➒ Calculate the NMR properties of hydrogen in HI, and HCl ➒ Use the prepared input files distributed in IS ➒ Take a look at the script run_respect.sh to see how ResPect run works. NOTE: analogous to ADF, there are two parts of ReSpect calculation β€’ SCF module-Calculation of unperturbed ground state MO Coefficient, in our case we started a guess SCF from (ksdkh2/pbe0) to (mdks/pbe0, i.e. Dirac Kohn Sham Implementation) β€’ CS module-Calculation of the chemical shift properties ➒ After the successful CS. Check the .out_cs file Look for the keyword β€œCHEMICAL SHIELDING FOR NUCLEUS # (your ATOM)” and below it is the β€œPrincipal values of the NMR shielding tensor ” where the PARA and DIA part shielding value(s) of the atom(s) sought. NOTE: this is the full Diract calculation, the SO effects are embedded in PARA and DIA, thus we can’t isolate the SO effects as opposed to the ZORA calculations. Exercise (ReSpect) STEP ReSpect Calc 11 Calc. Shielding Calc. Shielding Calc. Shielding Calc Shift Exp Shift (ref-TMS) Total Οƒref para dia Οƒi (para + dia) - [Ξ΄i (Οƒref - Οƒi) ] Ξ΄i *Ξ΄exp HCl_Full 31.41 0.45 HI_Full 31.41 13.25 ➒ After the successful of full Dirac calculation, compare some the values with the experiment, you should be able to complete this table: 12 Questions to Ponder ➒ from this activities, which method has the calculated value(s) that corroborates best with experimental 1H NMR shifts? On which system did the relativistic effect ➒ is choosing a computational method that contains approximations which lead to quicker and practical results worth it? Homework (optional task if you have free time hanging out in the wolf room) ➒ Try to calculate heavy metallic systems with experimental NMR datas, such as Platinum (195Pt) NMR Shift using Scalar ZORA, SO ZORA, and full Relativistic DFT calculations. ➒ Try this specie [Pt(NO2)4]2- with 195Pt experimental shift of – 2187 ppm. Use PtCl4 -2 for your reference. β€’ Use PBE0 functional for optimization and NMR shift calculations, and when it comes to the metal, use specific basis sets optimized for chemical shift calculations. Refer to the website of ADF and ResPect for the complete list. β€’ ADF (https://www.scm.com/doc/ADF/Input/Basis_sets_and_atomic_fragments.html#available-basis-sets) β€’ ReSpect (http://www.respectprogram.org/basis.html) Happy End Prepared by: Ben Joseph R. Cuyacot ➒ The Complete Table β€’ The SO effect as we can see here becomes more significant as the system involves atoms larger atoms (HCl to HI). The full Dirac still is the provides the best based on these geometries. 14 Answers Calc. Shielding Calc. Shielding Calc Shift Exp Shift (ref-TMS) Total Οƒref para dia Οƒi (para + dia) Ξ΄i (Οƒref - Οƒi) SO Ξ΄i + SO Ξ΄exp HCl_Scalar 31.35 0.83 30.54 31.38 -0.03 0.00 -0.03 0.45 HCl_SO 31.41 0.83 30.56 31.39 0.02 0.63 0.65 0.45 HI_Scalar 31.35 0.63 30.54 31.17 0.18 0.00 0.18 13.25 HI_SO 31.41 0.76 30.67 31.43 -0.02 9.41 9.39 13.25 Benzene_Scalar 31.35 -7.95 31.75 23.80 7.55 0.00 7.55 7.15 Benzene_SO 31.41 -7.95 31.75 23.80 7.61 -0.02 7.59 7.15 Calc. Shielding Calc. Shielding Calc Shift Exp Shift (ref-TMS) Total Οƒref para dia Οƒi (para + dia) (-) Ξ΄i (Οƒref - Οƒi) Ξ΄i + SO Ξ΄exp HCl_Full 31.41 2.41 29.29 31.70 0.29 0.29 0.45 HI_Full 31.41 16.13 28.63 44.76 13.35 13.35 13.25 Calc. Shielding Calc. Shielding