DFT calculations of NMR indirect spin spin coupling constants
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1 DFT calculations of NMR indirect spin spin coupling constants Dalton program system Program capabilities Density functional theory Kohn Sham theory LDA, GGA and hybrid theories Indirect NMR spin spin coupling constants Ramsey s second-order expression Computational aspects Calculation of spin spin coupling constants Comparison with high-level ab initio methods
2 Dalton program system Electronic-structure program system developed mostly in Scandinavia (Aarhus, Odense, Oslo and Stockholm) Computational methods Hartree Fock multiconfigurational self-consistent field (MCSCF) coupled-cluster theory density-functional theory Properties geometry optimizations and trajectories vibrational spectra (frequencies and intensities) excitation energies (triplet and singlet) static and dynamic polarizabilities and hyperpolarizabilities magnetic properties 570 licenses worldwide
3 Density functional theory The energy is expressed as a functional of the electron density r: The interaction with the external potential simple: By contrast, the kinetic energy T[ρ] and electron electron repulsion energy V ee [ρ] are difficult to calculate. However, an important, simple contribution to V ee [ρ] is the electron electron Coulomb repulsion energy:
4 Noninteracting electrons For a system of noninteracting electrons, we may write the density as In this case, the total energy may be written in the form where the kinetic energy is evaluated exactly as and the orbitals satisfy the eigenvalue equations
5 Kohn Sham theory
6 Kohn Sham and Hartree Fock theories With the density in the form, we have now expressed the DFT (Kohn Sham) energy in the form which is similar to that of the Hartree Fock energy The optimization may therefore be carried out in the same manner, replacing the Fock potential by the exchange correlation potential In practice, the evaluation of E xc [ρ] involves adjustable parameters; we may thus regard Kohn Sham theory as semi-empirical Hartree Fock theory.
7 LDA theory The true form of the exchange correlation functional is unknown and is presumably a complicated, nonlocal function of the electron density. However, reasonable results are obtained by assuming that the functional depends only locally on the density. In the local-density approximation (LDA), we use Dirac s (nonempirical) expression for the exchange energy For the correlation energy, the Vosko Wilk Nusair (VWN) functional (1980) is used; it is based on electron-gas simulations and contains three adjustable parameters.
8 GGA theory For quantitative results, it is necessary to include corrections from the gradient of the density such as With such gradient corrections included, we arrive at the generalized gradient-approximation (GGA) level of theory. The popular BLYP method uses Becke s gradient correction to the exchange (1988) one parameter adjusted to reproduce the exchange energy of the six noble-gas atoms the correlation correction due to Lee, Yang and Parr (1988) four parameters adjusted to reproduce the correlation energy of the helium atom
9 Hybrid DFT theories A successful Kohn Sham approach has been to include some proportion of the exact Hartree Fock exchange in the calculations: hybrid DFT In particular, the B3LYP functional contains the following contributions where the parameters have been fixed in a highly semi-empirical manner. B3LYP performs remarkably well in many circumstances for example, for the calculation of molecular structure, vibrational frequencies, atomization energies, and excitations. In the following, we shall concern ourselves with indirect NMR spin spin coupling constants.
10 Implementation of DFT In an existing ab initio code such as Dalton, Kohn Sham theory may be implemented as an extension to Hartree Fock theory. Wherever it appears, the Hartree Fock exchange contribution is replaced by a contribution from the exchange correlation functional: total energy Kohn Sham matrix linear Hessian transformations In hybrid theories, the Hartree Fock exchange is scaled rather than removed. The evaluation of the exchange correlation contributions involves an integration over all space: numerical scheme more convenient than an analytical one the space is divided into regions surrounding each nucleus within each such region, a one-dimensional radial and two-dimensional angular quadrature is carried out
11 Implementation in Dalton The implementation in Dalton is based on that in Cadpac it uses the same code for generating quadrature points and weights many of the same routines for the functionals Features of the implementation restricted Kohn Sham theory only full use of Abelian point-group symmetry parallelization over quadrature points generally contracted spherical-harmonic Gaussians The functional evaluation scales as KN 2, where K is the number of atoms and N the number of AOs.
12 Functionals: LDA, BLYP, B3LYP Features of Dalton DFT other functionals may be easily added Properties: total electronic energies molecular forces (and thus first-order geometry optimizations) multipole moments and other expectation values RPA singlet and triplet excitation energies NMR shieldings (with and without London orbitals) NMR indirect spin spin coupling constants static and dynamic polarizabilities magnetizabilities and molecular Hessians require some simple coding Planned for Dalton 2.0 release
13 NMR nuclear spin spin couplings In NMR, we observe direct and indirect couplings between the magnetic moments of paramagnetic nuclei Within the Born Oppenheimer approximation, these couplings may be calculated as second-order time-independent properties Because of the smallness of the moments (10-3 ) and their coupling to the electrons (10-5 ), these constants are extremely small: ~10-16 E h
14 NMR spectra
15 Nuclear magnetic perturbations Perturbed nonrelativistic electronic Hamiltonian: The nuclear magnetic vector potential gives rise to a first-order imaginary singlet perturbation and a secondorder real singlet perturbation The nuclear magnetic induction gives rise to a first-order real triplet perturbation.
16 Ramsey s expression The reduced spin-spin coupling constants are given by the paramagnetic spin-orbit (PSO) operator: the Fermi contact (FC) and spin-dipole (SD) operators:
17 Calculation of spin spin couplings In practice, second-order properties are not calculated from sum-overstates expressions Instead, the coupling constants are calculated from the expression Here λ S and λ T represent singlet and triplet variations of the electronic state, whose responses are obtained by solving the linear equations and similarly for the triplet perturbations. These response equations are solved iteratively, without constructing the Hessian matrices explicitly
18 NMR indirect nuclear spin spin couplings The calculation of NMR spin-spin coupling constants is a very demanding task: requires a highly flexible wave function to describe the triplet perturbations induced by the nuclear magnetic moments (triplet instabilities problems) large basis sets with a flexible inner core needed for each paramagnetic nucleus, we must solve at least nine response equations: 3 imaginary singlet perturbations 6 real triplet perturbations Standard ab initio approaches: RHF gives notoriously poor results CCSD is qualitatively but not quantitatively correct CCSD(T) not useful MCSCF and SOPPA are probably the most successful methods so far
19 DFT spin-spin methods compared Different DFT in HIII basis compared with RHF and full-valence CAS LDA represents a significant improvement on RHF Whereas RHF overestimates couplings, LDA underestimates them BLYP increases couplings and improves agreement somewhat Significant improvement at the B3LYP level
20 Contributions to spin-spin couplings In Ramsey s treatment, there are four contributions to the couplings. Usually, the most important is FC and the least important DSO. A priori, none of the couplings can be neglected.
21 Couplings compared (Hz)
22 Karplus relation The nuclear spin-spin couplings are extremely sensitive to geometries. Important special case: the Karplus relation between vicinal coupling constants and the dihedral angle φ between two CH bonds Below, we compare the B3LYP/HIII curve with an empirical curve
23 Conclusions For the calculation of indirect NMR spin spin coupling constants, LDA is a vast improvement on Hartree Fock theory no triplet-instability problems indirect nuclear spin spin couplings underestimated Some improvement observed at the GGA (BLYP) level B3LYP in semi-quantitative agreement with experiment typical errors less than 10% matches the best ab initio results (MCSCF, SOPPA, and CCSD) will presumably be no match for CCSDT may be applied to large systems cannot be systematically improved
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