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Prediction of NMR Parameters in the Solid-State

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Prediction of NMR Parameters in the Solid-State 1 JPO v. close to axial symmetry about Prediction of NMR parameters in the the P-O bond solid-state Jonathan Yates Materials Modelling Laboratory, Oxford Materials 2 JPSi 1 ‘z’ component slightly tilted from the internuclear vectors. Some deviation from axial symmetry Wednesday, 14 April 2010 Nuclear Magnetic Resonance Zero Field Applied Field B Population differences NMR signal is proportional to population difference •! For 1H in a 9.4 T field at 298 K For 1H in 9.4T at 298K "E = 2.65 # 10–25 J ω = γ B nupper/nlower = exp (–"E / kT) magnetogyric ratio nupper/nlower = 0.999935 fundamental nuclear constant Sensitivity is an issue! Need Large γ and/or B •! Populations differences in NMR are very small – sensitivity is always a concern •! Sensitivity is helped by higher magnetic field strengths and higher ! nuclei •! Need to generate a strong magnetic field (~2 to 28 T) •! Use a superconducting wire (~100 km for 850 MHz) NbTi 10 K 15 T ~£10 / m Nb3Sn 18.3 K 30 T ~£50 / m Common Isotopes ϒ Freq Abundance Isotope Spin 107 T-1 rad s-1 MHz @ 9.4T % 1H 1/2 26.75 400 100 13C 1/2 6.73 100.6 1.1 29Si 1/2 -5.32 79.6 4.7 31P 1/2 -10.84 162.1 100 Common Isotopes ϒ Freq Abundance Isotope Spin 107 T-1 rad s-1 MHz @ 9.4T % 1H 1/2 26.75 400 100 13C 1/2 6.73 100.6 1.1 29Si 1/2 -5.32 79.6 4.7 31P 1/2 -10.84 162.1 100 17O 5/2 -3.63 54.1 0.04 49Ti 7/2 -1.51 22.6 5.41 Solid-StateSuperconducting NMR Spectrometers magnets 600 MHz 14.1 T 400 MHz 9.4 T ~£800,000 ~£300,000 Image courtesy Sharon Ashbrook (St Andrews) Magnetic Shielding Flurbiprofen non-steriodal anti-inflamatory 6 (ω ωref ) 10 δiso = − × ωref chemical shift (ppm) δiso = σref σiso Each distinct C atom experiences a different magnetic field and − resonates at a unique frequency. Measure the change wrt a standard (for 13C this is liquid magnetic shielding tetramethylsilane) (from calc) 2 equivalently and by convention, within the 1st Brillouin planewave/pseudopotential formalism - it allows individ- Zone. ual operations to be performed in the most efficient basis, The problem of solving for an infinite number of elec- e.g. the kinetic energy operator in Fourier space, and a trons has become one of calculating for a finite number local potential in real space. of bands at an infinite number of k-points. However, physical properties are expected to be smoothly varying functions of k and hence many integrals can be well ap- D. The pseudopotential approximation proximated by a finite sampling of k. A common scheme for Brillouin Zone integration are the sets of regular in- The electrons in an atom can be divided into two types tegration grids introduced by Monkhorst and Pack. — core electrons and valence electrons. The core elec- trons are tightly bound to the nucleus, while the valence electrons are more extended. A working definition for 1. The supercell approximation core electrons is that they are the ones which play no part in the interactions between atoms, while the valence elec- The application of periodic boundary conditions forces trons dictate most of the properties of the material. It periodicity on the system studied. To address systems is common to make the frozen core approximation. The which do not have full three dimensional translational core electrons are constrained not to differ from their symmetry — for example the study of defects, impuri- free atomic nature when placed in the solid state envi- ties or even the interaction of molecules and surfaces, ronment. This reduces the number of electronic degrees the supercell approximation can be used. Aperiodic sys- of freedom in an all electron calculation. It is a very tems are approximated by enclosing the region of in- good approximation. A different, but physically related, terest in either bulk material (for a defect) or vacuum approach is taken in the pseudopotential approximation. (for a molecule) and then periodically repeating this cell Since, in an all electron calculation, the valence elec- throughout space (Figure??) The supercell must be large tron wavefunctions must be orthogonal to the core wave- enough for the fictitious interactions between neighboring functions they necessarily have strong oscillations in the cells to be negligible. region near the nucleus (see the all electron wavefunction in Figure 3). Given that a planewave basis set is to be used to describe the wavefunctions, these strong oscilla- C. The planewave basis set tions are undesirable — requiring many plane waves for an accurate description. Further, these oscillations are of In order to solve the eigenvalue problem of Equation 2 very little consequence for the electronic structure in the numerically the eigenstates must be represented by some solid, since they occur close to the nucleus, and interact basis set. While there are many possible choices, the one little with the neighboring atoms. In the pseudopotential made here is to use planewaves as the basis. There are approach only the valence electrons are explicitly consid- many advantages in the use of planewaves: they form ered, the effects of the core electrons being integrated a mathematically simple basis, they naturally incorpo- within a new ionic potential. The valence wavefunctions rate periodic boundary conditions, and, perhaps most need no longer be orthogonal to the core states, and so importantly planewave calculations can be taken system- the orthogonality oscillations disappear, hence far fewer atically to convergence as a function of the size of the plane waves are required to describe the valence wave- basis. The Kohn-Sham eigenstates are expressed as, functions. n n i(k+G).r Ψk(r) = ck(G)e , (5) !G IV. CHEMICAL SHIELDING IN A PERIODIC SYSTEM where the sum is over all reciprocal lattice vectors G. To truncate the basis set the sum is limited to a set of When a sample of matter is placed in a uniform, exter- reciprocal lattice vectors contained within a sphere with Induced Current a radius defined by the cutoff energy, E : nal magnetic field electronic currents flow throughout the cut material. For an insulating, non-magnetic material only !2|k + G|2 the orbitalTom computeotio nthe ochemicalf the shiftselect we justro needns toco calculatentribut e to this cur- ≤ E . (6) the current induced by the external magnetic field 2m cut rent. The current j(r), produces a non-uniform induced magnetic field in the material, which is given by the Biot- Hence, the basis set is defined by the maximum kinetic Savart law as energy component it contains. Physical quantities can be Biot-Savart r r! converged systematically by increasing Ecut. (figure of 1 3 ! ! − Bin(r) = d r j(r ) × . (7) shielding convergence) A Fast-Fourier Transform (FFT) c " |r − r!|3 can be used to change the representation of the eigen- The chemical shielding tensor is defined as the ratio be- states from a sum of Fourier components to to a uniform tween this induced field, and the external applied field grid of points in the real-space unit cell. The use of nu- Obtain current within perturbation merically efficient FFTs is one key to the success of the Bin(r) = −!σ(r)Bext. (8) theory (linear response) (0) (1) 2 O = O + O + (B ) Bin = -σ B0 O note: σ is a rank 2 tensor Planewaves and Pseudopotentials To simulate periodic systems planewaves are a convenient choice: however describing the tightly bound core states, and oscillatory part of the valence states close to the nucleus are prohibitively expensive. We must approximate... Frozen Core Approximation • “Core” electrons taken from free atom fixed during calculation Pseudopotential Approximation • Valence electrons experience weak effective potential in the core region Note: Typically these two approximations are used together. But this does not have to be the case. Some codes can employ ʻself-consistentʼ pseudopotentials which allow the core states to ʻrelaxʼ to their specific environment. Atomic States Si Approximations Overcoming the previous approximations Frozen Core Approximation • Contribution of “core” electrons to shielding is not chemically sensitive 1s states in Carbon contribute ~200ppm in diamond, benzene, proteins ie core states contribute to shielding - but not shift. Pseudopotential Approximation • Use PAW method to fix-up valence wavefunction in the core region COMPUTATIONS OF MAGNETIC RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5 COMPUTATIONS OF MAGNETIC RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5 The shielding of the reference standard, σ iso, could be iso ref 800 35 The shielding of the referenceobtained standard, by performingσref , could a be calculation on800 the35 standard com- obtained by performing a calculation on the standard com- 30 pound itself, but this is often not convenient—for30 example, if 600 pound itself, but this is often not convenient—for example, if 600 the reference is a liquid or solution. For this reason, most stud- 25 H the reference is a liquid or solution. For this reason,iso most stud- 25 H C ies have obtained σref as the intercept of a line of unit gradient 400 ies have obtained σ iso as the intercept of a line of unit gradient 400 20 C F ref fitted to a graph of calculated shieldings against20 experimental 20 25F 30 35 Si fitted to a graph of calculated shieldingsshifts. against experimental 20 25 30 35 200 Si P shifts. 200 P 0 0 GIPAW-USP (ppm) 3.2 Core States and PseudopotentialsGIPAW-USP (ppm) COMPUTATIONS3.2 Core States OF MAGNETIC and Pseudopotentials RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5 −200 COMPUTATIONS OF MAGNETIC RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5 In the planewave-pseudopotential approach,−200 it is implicit that In the planewave-pseudopotentialiso approach, it is implicit that 400 The shielding of the reference standard, σ , couldthe be core electrons35 can be treated separately from the valence − ref 800 iso 400 −400 −200 0 200 400 600 800 obtained by performingthe core a electrons calculation can on the beThe standard treated shielding com- separately of the reference from standard, the valenceσref , could be − 800 35 obtained bystates performing in an a calculation atomic30 code.
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