A Short Introduction into Quantum Chemical Calculations of NMR and EPR Parameters Martin Kaupp Institut für Chemie, TU Berlin • Part I: Some basics on quantum chemical methods • Part II: Basics on perturbation theory methods for the quantum-chemical calculation of NMR and EPR parameters a) Rayleigh-Schrödinger perturbation theory b) choice of Hamiltonian c) nuclear shieldings (diamagnetic systems) d) electronic g-tensor e) hyperfine coupling f) (zero-field splitting) pNMR-ITN Workshop Mariapfarr, Austria, February 23, 2014
A comprehensive treatment, Theory and Applications Wiley-VCH 2004. With 36 chapters on methodology and applications.
The effective spin Hamiltonian ˆ H = E model structure simulation properties spin Hamiltonian parameters Applications to: -spectra assignment -structure elucidation -interpretation comparison with model and/or -dynamics homologous systems ( e.g. from literature, database)
Introduction (1) effective spin Hamiltonian provides link between magnetic resonance experiment and quantum mechanical treatment: H S g B S A I S D S I q I N N N , 1 I B I D K I N N N NM NM M N N , M in most cases, property may be expressed as second derivative: H ( , ) E a b ( , ; , , ) X a S I b B S I a b a b appropriately treated by second-order perturbation theory: 0 V ( , ) H H a b E
Introduction (2) -relation between effective-spin Hamiltonian and quantum-chemical treatment requires appropriate Hamiltonian that takes care of spins and magnetic fields starting point relativistic quantum mechanics (e.g. Dirac equation) often transformation to quasi-nonrelativistic formalism (e.g. Breit-Pauli Hamiltonian) we will start by ignoring relativistic effects and magnetic interactions initially nonrelativistic quantum mechanics for H 0 relativistic effects and magnetic interactions will be introduced later, mainly by perturbation theory
Part I: Some Basics on Quantum Chemical Methods
H 0 : Nonrelativistic Quantum Mechanics (1) The general (non-relativistic) Hamiltonian describing N electrons and M nuclei is given (in a.u.) by: 1 1 Z 1 Z Z 2 2 A A B H i A 2 2 M r r R i 1 , N A 1 , M i 1 , N A 1 , M i 1 , N j i A 1 , M B A A iA ij AB
H 0 : Nonrelativistic Quantum Mechanics (2) after Born-Oppenheimer approximation (electronic Schrödinger equation):* electronic wave function ˆ H E electron-electron electronic repulsion Hamiltonian 1 1 ˆ ˆ H h ( i ) 2 r i i j one-electron ij Hamiltonian 1 ˆ ˆ 2 h ( i ) ( i ) V ( i ) 2 operator of potential energy operator of kinetic energy (mainly el.-nuclear attraction) still missing: electronic spin (Pauli principle), *and time dependence removed
H 0 : Nonrelativistic Quantum Mechanics (3) energy expectation value: H E 0 variational principle: ~ ~ H ~ E E ~ ~ 0 0 exact ground-state solution provides lower bound to energies of approximate solutions!
H 0 : Nonrelativistic Quantum Mechanics (4) in the absence of electron-electron interactions: ( 1 , 2 , 3 , 4 ,....... ) ( 1 ) ( 2 ) ( 3 ) ( 4 )..... ( ) n n 1 2 3 4 n Hartree product, independent-particle model, violates Pauli principle, no correlation. ( 1 , 2 , 3 , 4 ,....... n ) ( 1 ) ( 2 ) ( 3 ) ( 4 )..... ( n ) 1 2 3 4 n Slater determinant (antisym. linear combination of product wavefunctions) - i (i) : one-electron wave functions (molecular orbitals) -still no Coulomb correlation but Pauli exchange ~ insert as into variationa l principle ! Hartee-Fock method
H 0 : Nonrelativistic Quantum Mechanics (5) take trial wave function with free parameters q and minimize energy:* ~ ~ ~ H *Lagrange multipliers a E to account for constant 0 0 ~ ~ N(electrons), and for q q orthogonality of MOs for example: single Slater determinant Hartree-Fock method f ( 1 ) h ( 1 ) v ( 1 ) f ( 1 ) ( 1 ) ( 1 ) HF a a a 1 Z 2 A h ( 1 ) f depends on coordinates of 1 2 r A iA all electrons (via v HF ) ( 1 ) v J K iterative solution HF b b b 2 “self - consistent field” (SCF) method 1 J ( 1 ) dx ( 2 ) r b 2 b 12 i : molecular orbitals * 1 K ( 1 ) ( 1 ) dx ( 2 ) r ( 2 ) ( 1 ) J i : Coulomb op., K i : exchange op. b a 2 b 12 a b
H 0 : Nonrelativistic Quantum Mechanics (6) algebraic expansion of i into basis set: Roothan-Hall method matrix operations, linear algebra c ( A ) i ij j j optimizable parameters!!! : atomic orbital basis function with center on atom A MO expanded in linear combination of AOs MO-LCAO method typical basis sets: 2 r r A f ( r , , ) e B g ( r , , ) e Gaussian-type orbitals (GTO) Slater-type orbitals (STO) integrals more convenient fullfil cusp condition others: plane waves, numerical AOs, muffin- tin orbitals, etc……* *also for DFT etc……..
H 0 : Nonrelativistic Quantum Mechanics (7) Hartree-Fock method accounts for Pauli exchange but not for Coulomb correlation E HF ~ E (even with infinite basis set!) 0 0 ~ HF E E E Löwdin definition of correlation energy corr 0 0 better methods (e.g. post-Hartree-Fock or DFT), to account for E corr post-HF methods: perturbation theory (MPn) size-consistent, not variational configuration interaction (CI) not size-consistent,* variational coupled-cluster theory (CC) size-consistent, not variational *in truncated form
A Hierarchy of post -Hartree-Fock Theories Semi-Empirical Density MO-Theory Functional Hartree-Fock (HF), SCF, MO-LCAO Theory ~ ( ) 1 ( )...... 2 ( n 1 ) ( ) n (n 3 -n 4 ) 1 2 n 1 n n 4 Configuration Interaction (CI) Coupled Cluster Methods (CC) Perturbation Theory (MBPT) ~ ~ ( C C r r C rs rs ......) e T (e.g. CCSD: n 6 , 0 0 a a ab ab 0 (e.g. MP2: n 5 , MP3: n 6 , MP4: n 7 ) r r s ( T T T CCSD(T): n 7 ) ........) (e.g. SDCI: n 6 ) 1 2 “Dressed CI Methods” CPF, ACPF, MCPF, CEPA typically n 6 Multiconfiguration SCF (MCSCF, CASSCF, GVB) Multireference Multireference Multireference Perturbation Theory Configuration Interaction Coupled Cluster Theory (CASPT2, MR-MP2, MR-MPn?) (e.g. MR-SDCI, MR-ACPF) n m : formal scaling factors relative to system size n. Note that linear pre-factors are also important
Density Functional Theory (1) basic idea: use r (r) instead of the more complicated y (r1,r2,….rn): Hohenberg-Kohn Theorem (1963): E = E [ r ] E [ r ] = T [ r ] + V ne [ r ] + V ee [ r ] = r ( r ) ( r ) d r + F HK [ r ] Hohenberg-Kohn functional F HK [ r ] = T [ r ] + V ee [ r ] Kohn-Sham Method (1964): exchange-correlation F [ r ] = T s [ r ] + J [ r ] + E xc [ r ] functional of the KS method E xc [ r ] = T [ r ] - T s [ r ] + V ee [ r ] - J [ r ]
Density Functional Theory (2) Kohn-Sham equations: f ( 1 ) ( 1 ) ( 1 ) f ( 1 ) h ( 1 ) v ( 1 ) a a a xc while formally similar to HF-SCF equations, KS method incorporates electron correlation via (unknown) v xc approximations to E xc and xc : E xc r ]: local density approximation, LDA (e.g. SVWN, X ) E xc r, r ]: gen. gradient approximation, GGA (e.g. BLYP, BP86, PW91, PBE,….) E xc r, r, 2 r, t ]: meta-GGA (e.g. PKZB, FT98,…..) hybrid functionals (with exact exchange) (e.g. B3LYP, BHPW91, mPW1,…) further: OEP, exact- exchange functionals….
Density Functional Theory (3) See also: https://sites.google.com/site/markcasida/dft Local Density Approximation (LDA) 1/3 3 3 r r LDA LDA 4/3 r r r E ( ) d - d (r) x x 4 E c for homogeneous electron gas not known analytically but from very accurate fits to Quantum Monte Carlo simulations Generalized Gradient Approximation (GGA) r r r GGA LDA GGA r r r r E ( ) F ( ), ( ) d xc xc xc (different functionals have been constructed). Meta-Generalized Gradient Approximation (mGGA) r r r t mGGA mGGA 2 r r r r E ( ), ( ), ( ), ( ) d r xc xc 1 occ 2 t r r ( ) ( ) i 2 i
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