Energy Decomposition Analysis Introduction to Computational Quantum Chemistry Energy Decomposition Analysis Deformation density π›»πœŒ π‘Ÿ = 𝜌 π‘šπ‘œπ‘™ π‘Ÿ βˆ’ ෍ 𝑖=1 𝑁 π‘Žπ‘‘ πœŒπ‘– π‘Žπ‘‘ π‘Ÿ Positive values describe the point of density accumulation in the molecule. When the molecule is formed from atoms the density flows from the area of negative value towards the area of positive value. Formation of chemical bond in H2 –  based picture H H 1. Start from promolecular state (atom/fragments) H=1s2 H=1s2 H H H2 HH  22  Deformation density (Differential density) (1) qualitative data by inspection of the sign of : negative (outflow), positive (inflow) of density due to bond formation Extended Transition State Method Ziegler-Rauk : Bond formation in three steps A system A1. β€”B is cut into 2 (or more) fragments along an interesting bond: Aβ€”B β†’ A and B. The fragment are then overlapped from infinite distance to final position, i.e. form a superposition of ρA + ρB. The electrostatic attraction between the fragments is calculated quasi-classically (βˆ†Eelstat). The2. wavefunctions are allowed to overlap. The resulting wavefunction has to be antisymmetrized which increases the energy (βˆ†EPauli). The orbitals of the fragments are allowed to relax and3. electrons get redistributed to achieve the electronic state of the complete system. This lowers the energy again. This step involves mixing of orbitals. (βˆ†E= βˆ†Eorb) β€’ Decomposition of binding energy into different contributions. β€’ βˆ†Eint: Total intrinsic interaction energy between two (or more) fragments. β€’ βˆ†Eelstat: Attractive, quasi-classical electrostatic interaction between the electrons and the nuclei. β€’ βˆ†EPauli: Repulsive energy between electrons of the same spin. Comes from the antisymmetrization of the wavefunction. β€’ βˆ†Eorb: Comes from the relaxation of the fragment orbitals (mixing). Most interesting contribution to βˆ†Eint for a chemist. βˆ†Eorb can be decomposed into different contributions (Οƒ, Ο€, ...) via ETS-NOCV Energy components The Natural Orbitals for Chemical Valence (NOCV) NOCV’s also decompose the deformation density : useful qualitative data by inspection of the sign of : negative (outflow), positive (inflow) of density οƒ₯ ο€½ ο€½ M 1k 2 kk )r(ψv)r(ρ NOCV’s diagonalize the deformation density matrix: Mivi ,1=;ii CPC  where P=P-P0 , density matrix of the combined molecule, P0- density matrix of the considered molecular fragments. οƒ₯οƒ₯ ο€½ο€½ ο€­  2/ 1 2/ 1 22 )()]()([)( M k k M k kkk rrrvr  NOCV’s are in pairs: In short: The NOCV observes the electron flow when overlapping the fragments to come the final electron distribution. Each NOCV has a eigenvalue v assigned to it which shows how many electron are being moved into or out of the NOCV. The NOCVs are paired according to their eigenvalues and are then called NOCV pairs. NOCV pairs can be superimposed and are then called deformation densities. These are used to assign contributions of the orbital term to certain orbital interactions of the fragments in a compact way. In the deformation densities, the electrons flow from red β†’ blue -De=Etotal= Eprep + EintETS: A combination of ETS and NOCV - (ETS-NOCV) Eelstat + EPauli + Eorb Ξ”πœŒ 𝑁𝑂𝐢𝑉 Ξ”πœŒ1 + Ξ”πœŒ2 + Ξ”πœŒ3 + … Δ𝐸𝑖𝑛𝑑 𝐸𝑇𝑆 E1 + E2 + E3 +… ETS-NOCV Amsterdam Density Functional Very fast codeβ€’ Usesβ€’ Slater-type basis functions Functionalities:β€’ Optimizations Response properties (NMR, EPR, UV-VIS, IR, MΓΆssbauer..) NOCV EDA COSMO model of solvation ZORA scalar and spin-orbit relativistic approach Running ADF jobs β€’ Input: keywords in blocks β€’ adf {-n nproc} < input.adf > output.out β€’ nmr {-n nproc} < input_nmr.adf > output_ nmr.out β€’ INFINITY takes care of the number of CPUs β€’ Tape files: binaries containing the orbitals β€’ http://www.scm.com/Doc/Doc2014/ADF/ADFUsersGuide/page262.ht ml#keyscheme%20SAVE Advantages β€’ Many quantum-chemical tools implemented β€’ Relativistic effects included via ZORA Easy and intuitive syntax β€’ Good powerful GUI (ADFView) Disadvantages β€’ Slow geometry optimization β€’ Sometimes cryptic error messages β€’ Sometimes serious convergence problems Task β€’ Perform EDA with ETS-NOCV for NH3-BH3 β€’ Use the prepared input files distributed in IS. β€’ Three input files β€’ For fragment one NH3 β€’ For fragment two BH3 β€’ For the system NH3-BH3 β€’ Run adf for fragment 2. Rename the TAPE21 file to TAPE21 t21.frag2 β€’ Run adf for fragment 1. Rename the TAPE21 file to TAPE21 t21.frag1 β€’ Run adf for the system Relativistic calculation Why relativistic quantum chemistry In heavy elements the inner shellβ€’ elctrons move with a speed comparable with speed of light. So the core electrons of heavy atoms show sizable relativistic effect.β€’ For accurate predictions of various properties of chemical systemβ€’ containing heavy elements. Properties like NMR are very sensitive to thisβ€’ j mrel= π‘š π‘œ 1βˆ’ 𝜈2 𝐢 2 ෑ𝐻 π‘›π‘Ÿπ‘’π‘™ = ෍ 𝑖=1 Ƹ𝑝𝑖 2 2π‘š + ෍ 𝑖 𝑉 𝑁𝑖 + 𝑉𝑒 𝑒 + 𝑉𝑛 𝑒𝑐 ΰ·œπ‘π‘– 2 2π‘š is the non relativistic kinetic energy operator The Dirac equation Quantum mechanics Relativity The Dirac equation ෑ𝐻 𝐷𝐢𝐡 = ෠𝑃 + ෍ 𝑖=1 𝑁 (𝑐𝛼𝑖 Ƹ𝑝𝑖 + 𝛽𝑖 𝑐2) + 𝑉𝑁𝑖 + 𝑉𝑒 𝑒 + ෍ 𝑗>𝑖=1 𝑁 ෑ𝐻𝑖𝑗 𝐡 + 𝑉𝑛 𝑒𝑐 ෠𝑃 + Οƒ π‘₯ = 0 1 1 0 , Οƒy = 0 βˆ’π‘– 𝑖 0 , Οƒz = 1 0 0 βˆ’1 The main difference here is kinetic energy operator . Its takes into account the relativistic increase of electron mass due to high velocities, it includes the electron’s rest mass energy, it incorporates the electron spin also causes the spin orbit coupling. Here, Ξ± and Ξ² are 4 X 4 matrices is written in terms of the well known Pauli spinmatrices Οƒs. Breit interaction are relativistic corrections to Vee P+ is the projection operator Two-component relativistic methods β€’ High computational cost of four-component relativistic calculations has motivated the development of computationally less demanding two-component Hamiltonians. β€’ Two-component relativistic Hamiltonians (involving only positiveenergy orbitals) : pseudopotential and all-electron methods ZORA: accurate and efficient relativistic DFT 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. Spin Orbit Coupling can be included selfβ€’ -consistently All electron relativistic basis sets for all elementsβ€’ Available for most spectroscopic propertiesβ€’ Include relativistic effects on structure and reactivityβ€’ From Dirac to ZORA equation 𝑉 𝑐(𝜎. 𝑝) 𝑐(𝜎. 𝑝) 𝑉 βˆ’ 2𝑐2 . πœ™ πœ’ = 𝐸 πœ™ πœ’ 𝐻 π‘§π‘œπ‘Ÿπ‘Ž πœ™ π‘§π‘œπ‘Ÿπ‘Ž = 𝐸𝑧 π‘œπ‘Ÿπ‘Ž πœ™ π‘§π‘œπ‘Ÿπ‘Ž Unitary transformation Four-component Dirac Hamiltonian Two-component zeroth order regular approximation 𝐻 π‘§π‘œπ‘Ÿπ‘Ž = 𝑉 + 𝑝 𝑐2 2𝑐2 βˆ’π‘‰ 𝑝 + 𝑐2 2𝑐2 βˆ’π‘‰ 2 𝜎. (𝛻𝑉 Γ— 𝑝) Calculate the NMR properties of hydrogen in HI andβ€’ HCl Use the prepared input files distributed in ISβ€’ Write the molecular geometry in same format as .xyz (Γ…)β€’ Bond lengths:β€’ HCl(scalar): 1.276930 HI(scalar) : 1.606797 HCl(SO): 1.276467 HI(SO): 1.609681 For relativity useβ€’ ZORA Scalar and ZORA Spin-Orbit Approximations Compare the Experimental, Nonrelativistic, ECP, and Twoβ€’ component approach Chemical shielding for benzeneβ€’ 1H nuclei: NMR calculations ReSpect There are two parts of ReSpect calculation β€’ SCF module- Calculation of unperturbed ground state MO Coefficient. β€’ MAG module- Calculation of the magnetic properties . β€’ Each part requires a specific input file (.inp and .M) List of properties EPR β€’ G-tensor β€’ Hyperfine coupling tensor NMR β€’ NMR shielding tensor β€’ Nuclear spin-rotation coupling tensor β€’ Indirect nuclear spin-spin coupling tensor Functional: PBE, BP86, PBE0 etc(hybrid only for epr) Solvent: PCM model Calculate the NMR chemical shielding of H in HI usingβ€’ ReSpect Use the prepared input files distributed in IS.β€’ Three input file required.β€’ Initial guess,β€’ guess.inp Run using the commandβ€’ respect --np=4 --inp=guess Restart the calculation atβ€’ 4 component level from the initial guess respect --np=4 --inp=4c --restart=guess Calculate the NMR parameters using .M file.β€’ respect --np=4 --inp=4c --M=CS